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ps:: 0.0 471.37862 lineto ?ps:: 0.0 473.49442 1.71513 475.20956 3.83093 475.20956 curveto ps:: 464.17508 475.20956 lineto Ips:: 466.29086 475.20956 468.00601 473.49442 468.00601 471.37862 curveto ps:: 468.00601 0.0 lineto ps:: 467.00974 0.0 lineto ps:: 467.00974 471.37862 lineto Ips:: 467.00974 472.94418 465.74063 474.21329 464.17508 474.21329 curveto ps:: 3.83093 474.21329 lineto Cps:: 2.26538 474.21329 0.99628 472.94418 0.99628 471.37862 curveto ps:: 0.99628 0.0 lineto ps:: closepath ,ps:: gsave pgffc pgfeofill grestore newpath ps:: restore ps:: save ps:: /pgffc{0.95 setgray}def !ps:: 0.0 .pgfsetfillopacityalpha ps:: 0.99628 0.0 moveto ps:: 0.99628 471.3786 lineto Cps:: 0.99628 472.94417 2.26538 474.21327 3.83093 474.21327 curveto ps:: 464.17506 474.21327 lineto Hps:: 465.74062 474.21327 467.00972 472.94417 467.00972 471.3786 curveto ps:: 467.00972 0.0 lineto ps:: closepath *ps:: gsave pgffc pgffill grestore newpath ps:: restore ps:: save !ps:: 1.0 .pgfsetfillopacityalpha ps:: save 0ps:: [1.0 0.0 0.0 1.0 12.33499 8.78683 ] concat ps:: pgfs ps::[end]color push gray 02Vps:SDict begin H.S endps:SDict begin 13.6 H.A endSps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.2) cvn /DEST pdfmark endcolor push gray 0jSolution: color popLetfBX?ws 3 rm-lmr10= EfA}[ 3 rm-lmbx10}pz@E35rm-lmr8@1B;1E?B;1}pȮCd:JTEgBΕ= EfA}pzCjBEgFbCdmbjv@=1(P,whereeac!h}pzCjpE2 KAzCn NcanbMewrittenincoordinates}pzCjp: z= -u lmex10hËBpz@1:jCjB;1pz@2:jCjB;E?B;pzCn:jjFyHiK2,B:ps:SDict begin H.S endps:SDict begin 13.6 H.A endSps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.3) cvn /DEST pdfmark endps:SDict begin H.S endps:SDict begin 13.6 H.A endJps:SDict begin [/View [/XYZ H.V]/Dest (remark.1.0.2) cvn /DEST pdfmark end color push gray 0jRemark1.0.2:# color popProMofqxidea:fforsomexedBkɷwith2 EBkbEBn,|considerqxthepairs(Bpz@1:jCjB;1pzCk:jj#) E2KAz@2.rLetting Bjrangeo!ver 1\3EBjEBd yieldsBdpMoin!tsoftheform(Bx;1yd)\3E2KAz@2,csoconstruct anin!terpMolating polynomial such thatBf-(Bx)7=By%foreac!h tuple.-ThenBf-(Bx)EBy%vdDanishes atev!erysuc!hftuple.Doing&Pthisfor&Oeac!hBk~(keepingthe&OrstcoMordinatealwaysof&OtheformBpz@1:jCj PandlettingthesecondcoMordinatevdDary)yieldsBnE1polynomialsinBkX?[Bxz@1B;1xȮCk#]:E:Bk[Bxz@1B;1E?B;1xzCnP],thenaddinginthefpMolynomialBp(Bx) =C0Y 7ʍ{1Cj(BxnEBpz@1:jCj)fyieldsasystemthevdDanishespreciselyonEf }pzCjEg.Sz䍍 vps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save #ps:: 0 0.87 0.68 0.32 setcmykcolor ps:: 5.61972 3.10226 moveto ;ps:: 5.38145 3.14342 5.3347 2.9173 5.43457 2.83807 curveto ;ps:: 5.53458 2.75871 5.9751 2.83504 5.9751 2.83504 curveto ps:: 10.69238 5.82552 10.8247 6.08392 10.8247 6.08392 curveto >ps:: 10.5517 6.00443 10.4573 5.80759 10.57538 5.81638 curveto ps:: 8.95343 5.26703 moveto ;ps:: 9.10353 5.32288 9.2618 5.62242 9.2618 5.62242 curveto =ps:: 8.86557 5.46384 8.80365 5.21133 8.95343 5.26703 curveto ps:: 7.32346 4.70905 moveto ;ps:: 7.4614 4.7649 7.55548 5.07307 7.55548 5.07307 curveto ;ps:: 7.2764 4.92348 7.18536 4.6532 7.32346 4.70905 curveto ps:: 5.52835 3.87196 moveto ;ps:: 5.58722 3.81293 5.68996 3.9043 5.68996 3.9043 curveto =ps:: 5.84854 4.05696 5.91335 4.33893 5.91335 4.33893 curveto =ps:: 5.62532 4.18321 5.46962 3.93054 5.52835 3.87196 curveto ps:: 0.28142 0.1518 moveto ps:: 0.72772 0.16045 lineto =ps:: 0.72772 0.16045 1.68256 1.19595 3.03905 1.83842 curveto =ps:: 4.39555 2.48108 5.88565 2.14183 6.09113 2.67725 curveto =ps:: 6.09113 2.67725 5.36833 2.60588 5.32368 2.92737 curveto ps:: 11.38336 5.77429 11.66884 5.97063 12.142 5.75655 curveto =ps:: 12.142 5.75655 12.68623 5.8727 13.14117 6.08665 curveto @ps:: 13.59659 6.30092 13.66507 6.37485 13.61276 6.53201 curveto @ps:: 13.61276 6.53201 12.61421 6.31517 11.21867 5.85126 curveto =ps:: 9.82362 5.38737 9.04927 5.11423 7.49675 4.43332 curveto =ps:: 5.94423 3.75275 3.62906 2.48442 2.49913 1.77747 curveto ;ps:: 1.36955 1.07065 0.28142 0.1518 0.28142 0.1518 curveto ps:: 2.50331 1.87411 moveto ps:: 9.17027 5.28336 10.31232 5.6402 11.65985 6.05081 curveto =ps:: 13.0077 6.46127 13.6054 6.55968 13.6054 6.55968 curveto >ps:: 13.6054 6.55968 13.59659 6.80965 13.09685 6.7562 curveto @ps:: 12.59709 6.70258 11.91876 6.26524 11.91876 6.26524 curveto >ps:: 12.16888 6.48814 12.16888 6.6757 12.16888 6.6757 curveto >ps:: 11.14221 6.40797 10.92809 6.14908 10.77672 5.926 curveto @ps:: 10.62451 5.70308 10.50883 5.73878 10.50883 5.73878 curveto >ps:: 10.39264 5.81892 10.49971 6.03305 10.7852 6.1313 curveto >ps:: 11.07098 6.22923 11.0532 6.57744 11.0532 6.57744 curveto ?ps:: 10.73174 6.26524 10.14285 6.01529 9.76808 5.88135 curveto =ps:: 9.39316 5.74727 9.34853 5.65828 9.34853 5.65828 curveto =ps:: 9.08975 5.11389 8.69691 5.09596 8.83966 5.36353 curveto =ps:: 8.98256 5.63155 9.29506 5.68532 9.29506 5.68532 curveto :ps:: 9.43782 5.8279 9.49126 6.1758 9.49126 6.1758 curveto ps:: 9.50919 5.89928 10.6871 6.38127 11.13356 6.76515 curveto ps:: 11.07082 6.36348 lineto @ps:: 11.07082 6.36348 11.63312 6.66705 12.32924 6.86342 curveto ps:: 12.222 6.55984 lineto ;ps:: 12.222 6.55984 13.31958 7.04181 13.659 6.6757 curveto =ps:: 13.99808 6.30988 12.84673 5.81 12.31097 5.67636 curveto ?ps:: 12.31097 5.67636 12.5254 5.56003 12.20409 5.23009 curveto >ps:: 11.88292 4.89993 11.01704 4.8017 11.01704 4.8017 curveto ?ps:: 11.01704 4.8017 11.24062 4.46262 10.59796 4.18576 curveto ;ps:: 9.9553 3.9091 9.34853 3.93597 9.34853 3.93597 curveto ;ps:: 9.34853 3.93597 9.4023 3.48056 8.7419 3.31989 curveto =ps:: 8.08115 3.15941 7.57262 3.23972 7.57262 3.23972 curveto =ps:: 7.57262 3.23972 7.59038 2.92737 7.16183 2.76686 curveto =ps:: 6.73346 2.60603 6.13562 2.65982 6.13562 2.65982 curveto =ps:: 6.13562 2.65982 6.09113 2.25815 5.05579 2.15094 curveto ps:: gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore ps:: newpath ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]ps:SDict begin H.S endps:SDict begin 13.6 H.A endSps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.4) cvn /DEST pdfmark endčcolor push gray 0jClaim: color popWithoutHNlossHOofgeneralit!yf,pweHOcanassumealloftherstcompMonen!tsn Bpz@1:jCj5WomڍCdU`jv@=1/jare܀distinct.-sps:SDict begin H.S endps:SDict begin 13.6 H.A endHps:SDict begin [/View [/XYZ H.V]/Dest (section*.1) cvn /DEST pdfmark end?gps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save ps:: pgfstr ps:: save ps:: /pgfsc{0 setgray}def $ps:: /pgffc{1 0.5 0 setrgbcolor}def ps:: 0.49814 pgfw ps:: save ps:: /pgfsc{0 setgray}def $ps:: /pgffc{1 0.5 0 setrgbcolor}def ps:: 0.49814 pgfw ps:: 217.18475 5.81653 moveto ps:: -217.18475 5.81653 lineto Eps:: -219.3857 5.81653 -221.16986 4.03236 -221.16986 1.83142 curveto ps:: -221.16986 -1.83142 lineto Hps:: -221.16986 -4.03236 -219.3857 -5.81653 -217.18475 -5.81653 curveto ps:: 217.18475 -5.81653 lineto Eps:: 219.3857 -5.81653 221.16986 -4.03236 221.16986 -1.83142 curveto ps:: 221.16986 1.83142 lineto Bps:: 221.16986 4.03236 219.3857 5.81653 217.18475 5.81653 curveto ps:: closepath ps:: -221.16986 -5.81653 moveto >ps:: gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore ps:: newpath ps:: restore ps:: save 3ps:: [1.0 0.0 0.0 1.0 -217.41231 -2.05896 ] concat ps:: pgfs ps::[end]ps: /pgfsc{}def/pgffc{}def ps: 0 setgray kcolor push gray 0󀇸]ec-lmr6Tjo1do:follo9wsqfrom"rotationofaxes"? color pop ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]q Wfefwillusethefollo!wingfact: nps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save ps:: save ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 0ps:: [1.0 0.0 0.0 1.0 221.668 80.65826 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 0ps:: [1.0 0.0 0.0 1.0 221.668 80.65826 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: save &ps:: -16322.99998 -16322.99998 moveto %ps:: 16322.99998 -16322.99998 lineto $ps:: 16322.99998 16322.99998 lineto %ps:: -16322.99998 16322.99998 lineto ps:: closepath ps:: 0.0 2.83466 moveto ps:: 0.0 158.48189 lineto >ps:: 0.0 160.04745 1.2691 161.31656 2.83466 161.31656 curveto ps:: 440.50136 161.31656 lineto Ips:: 442.06693 161.31656 443.33603 160.04745 443.33603 158.48189 curveto ps:: 443.33603 2.83466 lineto :ps:: 443.33603 1.2691 442.06693 0.0 440.50136 0.0 curveto ps:: 2.83466 0.0 lineto /ps:: 1.2691 0.0 0.0 1.2691 0.0 2.83466 curveto ps:: closepath ps:: clip ps:: newpath ps:: restore ps:: save ps:: /pgffc{0.25 setgray}def !ps:: 1.0 .pgfsetfillopacityalpha ps:: 0.0 2.83466 moveto ps:: 0.0 158.48189 lineto >ps:: 0.0 160.04745 1.2691 161.31656 2.83466 161.31656 curveto ps:: 440.50136 161.31656 lineto Ips:: 442.06693 161.31656 443.33603 160.04745 443.33603 158.48189 curveto ps:: 443.33603 2.83466 lineto :ps:: 443.33603 1.2691 442.06693 0.0 440.50136 0.0 curveto ps:: 2.83466 0.0 lineto /ps:: 1.2691 0.0 0.0 1.2691 0.0 2.83466 curveto ps:: closepath ps:: 0.0 2.83466 moveto ps:: 0.0 158.48189 lineto >ps:: 0.0 160.04745 1.2691 161.31656 2.83466 161.31656 curveto ps:: 440.50136 161.31656 lineto Ips:: 442.06693 161.31656 443.33603 160.04745 443.33603 158.48189 curveto ps:: 443.33603 2.83466 lineto :ps:: 443.33603 1.2691 442.06693 0.0 440.50136 0.0 curveto ps:: 2.83466 0.0 lineto /ps:: 1.2691 0.0 0.0 1.2691 0.0 2.83466 curveto ps:: closepath ,ps:: gsave pgffc pgfeofill grestore newpath ps:: restore ps:: save ps:: /pgffc{0.95 setgray}def !ps:: 0.8 .pgfsetfillopacityalpha ps:: 0.0 2.83466 moveto ps:: 0.0 158.48189 lineto >ps:: 0.0 160.04745 1.2691 161.31656 2.83466 161.31656 curveto ps:: 440.50136 161.31656 lineto Ips:: 442.06693 161.31656 443.33603 160.04745 443.33603 158.48189 curveto ps:: 443.33603 2.83466 lineto :ps:: 443.33603 1.2691 442.06693 0.0 440.50136 0.0 curveto ps:: 2.83466 0.0 lineto /ps:: 1.2691 0.0 0.0 1.2691 0.0 2.83466 curveto ps:: closepath *ps:: gsave pgffc pgffill grestore newpath ps:: restore ps:: save ps:: 3.9851 pgfw %color push rgb 0.5843 0.33725 0.43922ps:: 1.99255 161.31654 moveto ps:: 1.99255 0.0 lineto ps:: pgfstr color pop ps:: restore ps:: save !ps:: 1.0 .pgfsetfillopacityalpha ps:: save 0ps:: [1.0 0.0 0.0 1.0 11.33871 7.79056 ] concat ps:: pgfs ps::[end]color push gray 0k>ps:SDict begin H.S endps:SDict begin 13.6 H.A endSps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.5) cvn /DEST pdfmark endps:SDict begin H.S endps:SDict begin 13.6 H.A endKps:SDict begin [/View [/XYZ H.V]/Dest (theorem.1.0.3) cvn /DEST pdfmark end6f,color push gray 0jTheorem21.0.3k(LZagrange). color popGiv!enqgasetofBdpMointsqfEf(BxzCidB;1yzCi)(ZEgE.0CdH.0i@=1?withallBxzCiAdistinct,|thereexistsauniquepMolynomialoffdegreeBdinBf8cE2 BkX?[Bx]suc!hthat@~Bf 0(BxzCid)=ByzCi @forev!eryBi.ThisfcanbMeexplicitlygiv!enby(ꍍ@b~}BfF (Bx) =C1Cd ύC0X 7ʍbCi@=1OByzCi婊8 08 B8 Bfi8 @C+C0Yw򤍍 @0Fbdlmsy8FCmFCd#mF6@=Ci-(7jBxnEBxzCm5@ȉfe%O tBxzCiEnBxzCm\;e婊c1cCcCficAn+ B:, EquivdDalen!tlyf,there;isJB: tThenftheroMotsofBfz@1ffarepreciselytherstcomponen!tsofthepoin!tsBp. jcolor push gray 0{끀 color pop`DenefBfz@2ʧE2 BkX?[Bxz@1B;1xz@2]EBkX?[Bxz@1B;1E?B;1xzCnP]fb!yconsideringtheorderedpairsy"ṉ(Bxz@1B;1xz@2) =(Bpz@1:jCjB;pz@2:jCj)o1}B; 8`thenҙtakingҘtheuniqueLagrangein!terpMolatingpolynomial@.~ҘBfz@2satisfying@.~Bfz@2 (Bpz@1:jCj)TP=Bpz@2:jCjfor 6`allf1 EBjEBd.ThensetBfz@2ʧ: p=@f#~Bfz@2 '(Bxz@1)nEBxz@2ʧE2 BkX?[Bxz@1B;1xz@2].jcolor push gray 0{끀 color pop`DenefBfz@3ʧE2 BkX?[Bxz@1B;1xz@3]EBkX?[Bxz@1B;1E?B;1xzCnP]fb!yconsideringtheorderedpairsy"ṉ(Bxz@1B;1xz@3) =(Bpz@1:jCjB;pz@3:jCj)o1}B; 8`thenҙtakingҘtheuniqueLagrangein!terpMolatingpolynomial@.~ҘBfz@3satisfying@.~Bfz@2 (Bpz@1:jCj)TP=Bpz@3:jCjfor 6`allf1 EBjEBd.ThensetBfz@3ʧ: p=@f#~Bfz@3 '(Bxz@1)nEBxz@3ʧE2 BkX?[Bxz@1B;1xz@3].jcolor push gray 0{끀 color pop`E1/Con!tinuingfinthisw!ayfuptoBfzCnE2 BkX?[Bxz@1B;1xzCnP]yieldsasystemofBnpMolynomials.&_ps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save ps:: save ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 0ps:: [1.0 0.0 0.0 1.0 221.668 18.61699 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 0ps:: [1.0 0.0 0.0 1.0 221.668 18.61699 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: save &ps:: -16322.99998 -16322.99998 moveto %ps:: 16322.99998 -16322.99998 lineto $ps:: 16322.99998 16322.99998 lineto %ps:: -16322.99998 16322.99998 lineto ps:: closepath ps:: 0.0 3.83093 moveto ps:: 0.0 33.40305 lineto ps:: 10.69238 5.82552 10.8247 6.08392 10.8247 6.08392 curveto >ps:: 10.5517 6.00443 10.4573 5.80759 10.57538 5.81638 curveto ps:: 8.95343 5.26703 moveto ;ps:: 9.10353 5.32288 9.2618 5.62242 9.2618 5.62242 curveto =ps:: 8.86557 5.46384 8.80365 5.21133 8.95343 5.26703 curveto ps:: 7.32346 4.70905 moveto ;ps:: 7.4614 4.7649 7.55548 5.07307 7.55548 5.07307 curveto ;ps:: 7.2764 4.92348 7.18536 4.6532 7.32346 4.70905 curveto ps:: 5.52835 3.87196 moveto ;ps:: 5.58722 3.81293 5.68996 3.9043 5.68996 3.9043 curveto =ps:: 5.84854 4.05696 5.91335 4.33893 5.91335 4.33893 curveto =ps:: 5.62532 4.18321 5.46962 3.93054 5.52835 3.87196 curveto ps:: 0.28142 0.1518 moveto ps:: 0.72772 0.16045 lineto =ps:: 0.72772 0.16045 1.68256 1.19595 3.03905 1.83842 curveto =ps:: 4.39555 2.48108 5.88565 2.14183 6.09113 2.67725 curveto =ps:: 6.09113 2.67725 5.36833 2.60588 5.32368 2.92737 curveto ps:: 11.38336 5.77429 11.66884 5.97063 12.142 5.75655 curveto =ps:: 12.142 5.75655 12.68623 5.8727 13.14117 6.08665 curveto @ps:: 13.59659 6.30092 13.66507 6.37485 13.61276 6.53201 curveto @ps:: 13.61276 6.53201 12.61421 6.31517 11.21867 5.85126 curveto =ps:: 9.82362 5.38737 9.04927 5.11423 7.49675 4.43332 curveto =ps:: 5.94423 3.75275 3.62906 2.48442 2.49913 1.77747 curveto ;ps:: 1.36955 1.07065 0.28142 0.1518 0.28142 0.1518 curveto ps:: 2.50331 1.87411 moveto ps:: 9.17027 5.28336 10.31232 5.6402 11.65985 6.05081 curveto =ps:: 13.0077 6.46127 13.6054 6.55968 13.6054 6.55968 curveto >ps:: 13.6054 6.55968 13.59659 6.80965 13.09685 6.7562 curveto @ps:: 12.59709 6.70258 11.91876 6.26524 11.91876 6.26524 curveto >ps:: 12.16888 6.48814 12.16888 6.6757 12.16888 6.6757 curveto >ps:: 11.14221 6.40797 10.92809 6.14908 10.77672 5.926 curveto @ps:: 10.62451 5.70308 10.50883 5.73878 10.50883 5.73878 curveto >ps:: 10.39264 5.81892 10.49971 6.03305 10.7852 6.1313 curveto >ps:: 11.07098 6.22923 11.0532 6.57744 11.0532 6.57744 curveto ?ps:: 10.73174 6.26524 10.14285 6.01529 9.76808 5.88135 curveto =ps:: 9.39316 5.74727 9.34853 5.65828 9.34853 5.65828 curveto =ps:: 9.08975 5.11389 8.69691 5.09596 8.83966 5.36353 curveto =ps:: 8.98256 5.63155 9.29506 5.68532 9.29506 5.68532 curveto :ps:: 9.43782 5.8279 9.49126 6.1758 9.49126 6.1758 curveto ps:: 9.50919 5.89928 10.6871 6.38127 11.13356 6.76515 curveto ps:: 11.07082 6.36348 lineto @ps:: 11.07082 6.36348 11.63312 6.66705 12.32924 6.86342 curveto ps:: 12.222 6.55984 lineto ;ps:: 12.222 6.55984 13.31958 7.04181 13.659 6.6757 curveto =ps:: 13.99808 6.30988 12.84673 5.81 12.31097 5.67636 curveto ?ps:: 12.31097 5.67636 12.5254 5.56003 12.20409 5.23009 curveto >ps:: 11.88292 4.89993 11.01704 4.8017 11.01704 4.8017 curveto ?ps:: 11.01704 4.8017 11.24062 4.46262 10.59796 4.18576 curveto ;ps:: 9.9553 3.9091 9.34853 3.93597 9.34853 3.93597 curveto ;ps:: 9.34853 3.93597 9.4023 3.48056 8.7419 3.31989 curveto =ps:: 8.08115 3.15941 7.57262 3.23972 7.57262 3.23972 curveto =ps:: 7.57262 3.23972 7.59038 2.92737 7.16183 2.76686 curveto =ps:: 6.73346 2.60603 6.13562 2.65982 6.13562 2.65982 curveto =ps:: 6.13562 2.65982 6.09113 2.25815 5.05579 2.15094 curveto ps:: gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore ps:: newpath ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]'dps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save ps:: save ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 3ps:: [1.0 0.0 0.0 1.0 234.00299 140.08739 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 3ps:: [1.0 0.0 0.0 1.0 234.00299 140.08739 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: save &ps:: -16322.99998 -16322.99998 moveto %ps:: 16322.99998 -16322.99998 lineto $ps:: 16322.99998 16322.99998 lineto %ps:: -16322.99998 16322.99998 lineto ps:: closepath ps:: 0.0 0.0 moveto ps:: 0.0 276.34386 lineto ?ps:: 0.0 278.45964 1.71513 280.17479 3.83093 280.17479 curveto ps:: 464.17508 280.17479 lineto Ips:: 466.29086 280.17479 468.00601 278.45964 468.00601 276.34386 curveto ps:: 468.00601 0.0 lineto ps:: closepath ps:: clip ps:: newpath ps:: restore ps:: save .ps:: /pgffc{0.62746 0.76863 1 setrgbcolor}def !ps:: 1.0 .pgfsetfillopacityalpha ps:: 0.0 0.0 moveto ps:: 0.0 276.34386 lineto ?ps:: 0.0 278.45964 1.71513 280.17479 3.83093 280.17479 curveto ps:: 464.17508 280.17479 lineto Ips:: 466.29086 280.17479 468.00601 278.45964 468.00601 276.34386 curveto ps:: 468.00601 0.0 lineto ps:: 467.00974 0.0 lineto ps:: 467.00974 276.34386 lineto Ips:: 467.00974 277.90941 465.74063 279.17851 464.17508 279.17851 curveto ps:: 3.83093 279.17851 lineto Cps:: 2.26538 279.17851 0.99628 277.90941 0.99628 276.34386 curveto ps:: 0.99628 0.0 lineto ps:: closepath ,ps:: gsave pgffc pgfeofill grestore newpath ps:: restore ps:: save ps:: /pgffc{0.95 setgray}def !ps:: 0.0 .pgfsetfillopacityalpha ps:: 0.99628 0.0 moveto ps:: 0.99628 276.34386 lineto Cps:: 0.99628 277.90941 2.26538 279.17851 3.83093 279.17851 curveto ps:: 464.17506 279.17851 lineto Ips:: 465.74062 279.17851 467.00972 277.90941 467.00972 276.34386 curveto ps:: 467.00972 0.0 lineto ps:: closepath *ps:: gsave pgffc pgffill grestore newpath ps:: restore ps:: save !ps:: 1.0 .pgfsetfillopacityalpha ps:: save 0ps:: [1.0 0.0 0.0 1.0 12.33499 6.66562 ] concat ps:: pgfs ps::[end]color push gray 0(ps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.11) cvn /DEST pdfmark endcolor push gray 0jSolution: color popFforfnotationalpurpMoses,letEIB;1EVdenotethemapsinHilbert'sNullstellensatz,w!ethenhaveȍ(EI=nV|)(BI) =6Ep *6p 6 [BI:So w!e considerEV|(BI)yEyKAz@2B=KC,thevdDanishingloMcusoftheset!wo pMolynomials,whichyields thesystemf(]ӍtUBxz@3.EnBydz@6Q= 0Q獍tUBxyEnBydz@3Q= 0B:In$the$secondequation,Diw!ehave$(Bx1EBydz@2$)ByA=U0,Dhand$sinceKC[Bx;1yd]isanin!tegraldomain,Dhonetermfm!ustbMezero.|ps:SDict begin H.S endps:SDict begin 13.6 H.A endDps:SDict begin [/View [/XYZ H.V]/Dest (Item.1) cvn /DEST pdfmark endRLjcolor push gray 0{1. color pop`IfؘBy=^O0,%thenBxz@3P=0=,E)Bx^P=0,$andؘth!us(0B;10)^PE2V|(BI),i.e.ؘtheؙoriginiscon!tainedin`thisfvdDanishingloMcus.ps:SDict begin H.S endps:SDict begin 13.6 H.A endDps:SDict begin [/View [/XYZ H.V]/Dest (Item.2) cvn /DEST pdfmark end1=jcolor push gray 0{2. color pop`Otherwise, ifBxEBydz@2^=:.0,thenBx=Bydz@2$,withnofurtherconditionscomingfromtherst`equation.ȍCom!biningftheseconditions, 獍jBP:= n S(Btz@2B;1t)C.f f f !Bt E2KCGoQVkE V|(BI)B:) color pop ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]/color push gray 01L color popvgcolor push gray 0WExercises+-color push gray 0X4 color pop -ɍ냍color push gray 0ff color pop-ɍ냍color push gray 0f color popG color pop6@Oᎎҍᎎps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if endP ps:SDict begin H.S endps:SDict begin H.R endDps:SDict begin [/View [/XYZ H.V]/Dest (page.5) cvn /DEST pdfmark endɿ1덍color push gray 0iFC냍Tvgcolor push gray 0 =color push gray 0.6color push gray 0.6ٚ$_Ž color popcolor push gray 0LGcolor push gray 11 color popןٚ$ǎ_ color pop_ color pop3Exercises3]T color pop67냍Tcolor push gray 03]T color pop;+8 Tcolor push gray 03]T3]T color popG color pop{͍ps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save ps:: save ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 1ps:: [1.0 0.0 0.0 1.0 234.00299 51.4734 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 1ps:: [1.0 0.0 0.0 1.0 234.00299 51.4734 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: save &ps:: -16322.99998 -16322.99998 moveto %ps:: 16322.99998 -16322.99998 lineto $ps:: 16322.99998 16322.99998 lineto %ps:: -16322.99998 16322.99998 lineto ps:: closepath ps:: 0.0 3.83093 moveto ps:: 0.0 102.94684 lineto ps:: 468.00601 102.94684 lineto ps:: 468.00601 3.83093 lineto ;ps:: 468.00601 1.71513 466.29086 0.0 464.17508 0.0 curveto ps:: 3.83093 0.0 lineto 1ps:: 1.71513 0.0 0.0 1.71513 0.0 3.83093 curveto ps:: closepath ps:: clip ps:: newpath ps:: restore ps:: save .ps:: /pgffc{0.62746 0.76863 1 setrgbcolor}def !ps:: 1.0 .pgfsetfillopacityalpha ps:: 0.0 3.83093 moveto ps:: 0.0 102.94684 lineto ps:: 0.99628 102.94684 lineto ps:: 0.99628 3.83093 lineto =ps:: 0.99628 2.26538 2.26538 0.99628 3.83093 0.99628 curveto ps:: 464.17508 0.99628 lineto Cps:: 465.74063 0.99628 467.00974 2.26538 467.00974 3.83093 curveto ps:: 467.00974 102.94684 lineto ps:: 468.00601 102.94684 lineto ps:: 468.00601 3.83093 lineto ;ps:: 468.00601 1.71513 466.29086 0.0 464.17508 0.0 curveto ps:: 3.83093 0.0 lineto 1ps:: 1.71513 0.0 0.0 1.71513 0.0 3.83093 curveto ps:: closepath ,ps:: gsave pgffc pgfeofill grestore newpath ps:: restore ps:: save ps:: /pgffc{0.95 setgray}def !ps:: 0.0 .pgfsetfillopacityalpha ps:: 0.99628 3.83093 moveto ps:: 0.99628 102.94682 lineto ps:: 467.00972 102.94682 lineto ps:: 467.00972 3.83093 lineto Cps:: 467.00972 2.26538 465.74062 0.99628 464.17506 0.99628 curveto ps:: 3.83093 0.99628 lineto =ps:: 2.26538 0.99628 0.99628 2.26538 0.99628 3.83093 curveto ps:: closepath *ps:: gsave pgffc pgffill grestore newpath ps:: restore ps:: save !ps:: 1.0 .pgfsetfillopacityalpha ps:: save 0ps:: [1.0 0.0 0.0 1.0 12.33499 12.9049 ] concat ps:: pgfs ps::[end]color push gray 0 wherefBI= D 'Bxz@3.EnBydz@6$B;1xyEBydz@3T1EZ.yq Wfefha!veBP= EV|(BI),andsotakingtheidealgeneratedbyBP+yieldsy (CJEI=nVJ)(BI) =EI(BPV)=D 'ByEnBxz@2'N@E0uYE2KC[Bx;1yd] andfth!us6Ep g6p 6 [BI}D= D 'ByEnBxz@2'N@E-j.) color pop ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]color push gray 0 color popps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.12) cvn /DEST pdfmark endps:SDict begin H.S endps:SDict begin 13.6 H.A endLps:SDict begin [/View [/XYZ H.V]/Dest (exercise.1.0.6) cvn /DEST pdfmark end34color push gray 0jExercise1.0.6k(Gathmann1.22):# color popLetBXRE@[KAz@3B=kRbMetheunionofthethreecoMordinateaxes. ComputegeneratorsfortheidealBI(BX)andsho!wthatitcannotbMegeneratedb!yfewerthan3elemen!ts.,΍z䍍 vps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save #ps:: 0 0.87 0.68 0.32 setcmykcolor ps:: 5.61972 3.10226 moveto ;ps:: 5.38145 3.14342 5.3347 2.9173 5.43457 2.83807 curveto ;ps:: 5.53458 2.75871 5.9751 2.83504 5.9751 2.83504 curveto ps:: 10.69238 5.82552 10.8247 6.08392 10.8247 6.08392 curveto >ps:: 10.5517 6.00443 10.4573 5.80759 10.57538 5.81638 curveto ps:: 8.95343 5.26703 moveto ;ps:: 9.10353 5.32288 9.2618 5.62242 9.2618 5.62242 curveto =ps:: 8.86557 5.46384 8.80365 5.21133 8.95343 5.26703 curveto ps:: 7.32346 4.70905 moveto ;ps:: 7.4614 4.7649 7.55548 5.07307 7.55548 5.07307 curveto ;ps:: 7.2764 4.92348 7.18536 4.6532 7.32346 4.70905 curveto ps:: 5.52835 3.87196 moveto ;ps:: 5.58722 3.81293 5.68996 3.9043 5.68996 3.9043 curveto =ps:: 5.84854 4.05696 5.91335 4.33893 5.91335 4.33893 curveto =ps:: 5.62532 4.18321 5.46962 3.93054 5.52835 3.87196 curveto ps:: 0.28142 0.1518 moveto ps:: 0.72772 0.16045 lineto =ps:: 0.72772 0.16045 1.68256 1.19595 3.03905 1.83842 curveto =ps:: 4.39555 2.48108 5.88565 2.14183 6.09113 2.67725 curveto =ps:: 6.09113 2.67725 5.36833 2.60588 5.32368 2.92737 curveto ps:: 11.38336 5.77429 11.66884 5.97063 12.142 5.75655 curveto =ps:: 12.142 5.75655 12.68623 5.8727 13.14117 6.08665 curveto @ps:: 13.59659 6.30092 13.66507 6.37485 13.61276 6.53201 curveto @ps:: 13.61276 6.53201 12.61421 6.31517 11.21867 5.85126 curveto =ps:: 9.82362 5.38737 9.04927 5.11423 7.49675 4.43332 curveto =ps:: 5.94423 3.75275 3.62906 2.48442 2.49913 1.77747 curveto ;ps:: 1.36955 1.07065 0.28142 0.1518 0.28142 0.1518 curveto ps:: 2.50331 1.87411 moveto ps:: 9.17027 5.28336 10.31232 5.6402 11.65985 6.05081 curveto =ps:: 13.0077 6.46127 13.6054 6.55968 13.6054 6.55968 curveto >ps:: 13.6054 6.55968 13.59659 6.80965 13.09685 6.7562 curveto @ps:: 12.59709 6.70258 11.91876 6.26524 11.91876 6.26524 curveto >ps:: 12.16888 6.48814 12.16888 6.6757 12.16888 6.6757 curveto >ps:: 11.14221 6.40797 10.92809 6.14908 10.77672 5.926 curveto @ps:: 10.62451 5.70308 10.50883 5.73878 10.50883 5.73878 curveto >ps:: 10.39264 5.81892 10.49971 6.03305 10.7852 6.1313 curveto >ps:: 11.07098 6.22923 11.0532 6.57744 11.0532 6.57744 curveto ?ps:: 10.73174 6.26524 10.14285 6.01529 9.76808 5.88135 curveto =ps:: 9.39316 5.74727 9.34853 5.65828 9.34853 5.65828 curveto =ps:: 9.08975 5.11389 8.69691 5.09596 8.83966 5.36353 curveto =ps:: 8.98256 5.63155 9.29506 5.68532 9.29506 5.68532 curveto :ps:: 9.43782 5.8279 9.49126 6.1758 9.49126 6.1758 curveto ps:: 9.50919 5.89928 10.6871 6.38127 11.13356 6.76515 curveto ps:: 11.07082 6.36348 lineto @ps:: 11.07082 6.36348 11.63312 6.66705 12.32924 6.86342 curveto ps:: 12.222 6.55984 lineto ;ps:: 12.222 6.55984 13.31958 7.04181 13.659 6.6757 curveto =ps:: 13.99808 6.30988 12.84673 5.81 12.31097 5.67636 curveto ?ps:: 12.31097 5.67636 12.5254 5.56003 12.20409 5.23009 curveto >ps:: 11.88292 4.89993 11.01704 4.8017 11.01704 4.8017 curveto ?ps:: 11.01704 4.8017 11.24062 4.46262 10.59796 4.18576 curveto ;ps:: 9.9553 3.9091 9.34853 3.93597 9.34853 3.93597 curveto ;ps:: 9.34853 3.93597 9.4023 3.48056 8.7419 3.31989 curveto =ps:: 8.08115 3.15941 7.57262 3.23972 7.57262 3.23972 curveto =ps:: 7.57262 3.23972 7.59038 2.92737 7.16183 2.76686 curveto =ps:: 6.73346 2.60603 6.13562 2.65982 6.13562 2.65982 curveto =ps:: 6.13562 2.65982 6.09113 2.25815 5.05579 2.15094 curveto ps:: gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore ps:: newpath ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]ps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save ps:: save ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 2ps:: [1.0 0.0 0.0 1.0 234.00299 239.5664 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 2ps:: [1.0 0.0 0.0 1.0 234.00299 239.5664 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: save &ps:: -16322.99998 -16322.99998 moveto %ps:: 16322.99998 -16322.99998 lineto $ps:: 16322.99998 16322.99998 lineto %ps:: -16322.99998 16322.99998 lineto ps:: closepath ps:: 0.0 0.0 moveto ps:: 0.0 475.30186 lineto ?ps:: 0.0 477.41766 1.71513 479.13281 3.83093 479.13281 curveto ps:: 464.17508 479.13281 lineto Ips:: 466.29086 479.13281 468.00601 477.41766 468.00601 475.30186 curveto ps:: 468.00601 0.0 lineto ps:: closepath ps:: clip ps:: newpath ps:: restore ps:: save .ps:: /pgffc{0.62746 0.76863 1 setrgbcolor}def !ps:: 1.0 .pgfsetfillopacityalpha ps:: 0.0 0.0 moveto ps:: 0.0 475.30186 lineto ?ps:: 0.0 477.41766 1.71513 479.13281 3.83093 479.13281 curveto ps:: 464.17508 479.13281 lineto Ips:: 466.29086 479.13281 468.00601 477.41766 468.00601 475.30186 curveto ps:: 468.00601 0.0 lineto ps:: 467.00974 0.0 lineto ps:: 467.00974 475.30186 lineto Ips:: 467.00974 476.86743 465.74063 478.13654 464.17508 478.13654 curveto ps:: 3.83093 478.13654 lineto Cps:: 2.26538 478.13654 0.99628 476.86743 0.99628 475.30186 curveto ps:: 0.99628 0.0 lineto ps:: closepath ,ps:: gsave pgffc pgfeofill grestore newpath ps:: restore ps:: save ps:: /pgffc{0.95 setgray}def !ps:: 0.0 .pgfsetfillopacityalpha ps:: 0.99628 0.0 moveto ps:: 0.99628 475.30186 lineto Cps:: 0.99628 476.86743 2.26538 478.13654 3.83093 478.13654 curveto ps:: 464.17506 478.13654 lineto Ips:: 465.74062 478.13654 467.00972 476.86743 467.00972 475.30186 curveto ps:: 467.00972 0.0 lineto ps:: closepath *ps:: gsave pgffc pgffill grestore newpath ps:: restore ps:: save !ps:: 1.0 .pgfsetfillopacityalpha ps:: save 0ps:: [1.0 0.0 0.0 1.0 12.33499 6.66562 ] concat ps:: pgfs ps::[end]color push gray 0,u,ps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.13) cvn /DEST pdfmark endcolor push gray 0jSolution: color popClaim u:dBI(BX) =EhLBxz@2Bxz@3B;bxz@1Bxz@3B;xz@1Bxz@2VEi\B:q WfefcanwriteBX= BXz@1.E[nBXz@2E[BXz@3,fwherejcolor push gray 0{끀 color pop`ThefBxz@1-ffaxisisgiv!enbyBXz@1ʧ: p= BVn(Bxz@2Bxz@3) =,E)NBI(BXz@1)=EhLBxz@2Bxz@3QEi!#, jcolor push gray 0{끀 color pop`ThefBxz@2-ffaxisisgiv!enbyBXz@2ʧ: p= BVn(Bxz@1Bxz@3) =,E)NBI(BXz@2)=EhLBxz@1Bxz@3QEi!#,jcolor push gray 0{끀 color pop`ThefBxz@3-ffaxisisgiv!enbyBXz@3ʧ: p= BVn(Bxz@1Bxz@2) =,E)NBI(BXz@3)=EhLBxz@1Bxz@2QEi!#.Herefw!e'veused,forexample,that0\BI(BVn(Bxz@2Bxz@3)) =Qq Qfe| SEhB#Bxz@2Bxz@3FYEi.=EhLBxz@2Bxz@3QEib!yapplyingtheNullstellensatzandnotingthatEhBxz@2Bxz@3Ei'isradicalsinceitisgeneratedb!yasquarefreefmonomial.q Wfefthenha!ve8XBI(BX)S= BI(BXz@1.E[nBXz@2E[BXz@3)S= BI(BXz@1)nE\BI(BXz@2)E\BI(BXz@3)4S= Pq Pfen SBI(BXz@1)n+BI(BXz@2)+BI(BXz@3)foS= Qq Qfe{; SEhB#Bxz@2B;1xz@3$5Ei%D+nEhBxz@1Bxz@3EEi#fT+nEhBxz@1Bxz@2EEifpS= Qq QfeW SEhB#Bxz@2Bxz@3B;bxz@1Bxz@3B;xz@1Bxz@2SߝEiBsince>NEh݀qBaEi>+nEhBbEi= EhLBa;1bnEiS= EhLBxz@2Bxz@3B;bxz@1Bxz@3B;xz@1Bxz@2VEiZB;whereʒinthelastʑequalit!ywe'veagainusedthefactthatanʑidealgeneratedbysquarefreemonomialsfisradical.ps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.14) cvn /DEST pdfmark end!color push gray 0jClaim:U color popBI(BX)fcannotbMegeneratedb!y2orfewerelements.Let6BJl󞞩:=_WBI(BX)5andBRt:=_VBkX?[Bxz@1B;1xz@2B;xz@3],and6to!ward5acontradiction,suppMoseBJl=_VEhyBr";1sD&EiI.vMDene%n 3 eufm10m :Zp= EhLBx;1yd;z"tEi*andfaquotien!tmapBo:: BJDE!BJq=mBJ) color pop ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]/color push gray 01L color popvgcolor push gray 0WExercises+-color push gray 0X5 color pop -ɍ냍color push gray 0ff color pop-ɍ냍color push gray 0f color popG color pop6@Oᎎᎎps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if endP ps:SDict begin H.S endps:SDict begin H.R endDps:SDict begin [/View [/XYZ H.V]/Dest (page.6) cvn /DEST pdfmark endɿ1덍color push gray 0iFC냍Tvgcolor push gray 0 =color push gray 0.6color push gray 0.6ٚ$_Ž color popcolor push gray 0LGcolor push gray 11 color popןٚ$ǎ_ color pop_ color pop3Exercises3]T color pop67냍Tcolor push gray 03]T color pop;+8 Tcolor push gray 03]T3]T color popG color pop{sqps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save ps:: save ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 3ps:: [1.0 0.0 0.0 1.0 234.00299 126.62155 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 3ps:: [1.0 0.0 0.0 1.0 234.00299 126.62155 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: save &ps:: -16322.99998 -16322.99998 moveto %ps:: 16322.99998 -16322.99998 lineto $ps:: 16322.99998 16322.99998 lineto %ps:: -16322.99998 16322.99998 lineto ps:: closepath ps:: 0.0 3.83093 moveto ps:: 0.0 253.24313 lineto ps:: 468.00601 253.24313 lineto ps:: 468.00601 3.83093 lineto ;ps:: 468.00601 1.71513 466.29086 0.0 464.17508 0.0 curveto ps:: 3.83093 0.0 lineto 1ps:: 1.71513 0.0 0.0 1.71513 0.0 3.83093 curveto ps:: closepath ps:: clip ps:: newpath ps:: restore ps:: save .ps:: /pgffc{0.62746 0.76863 1 setrgbcolor}def !ps:: 1.0 .pgfsetfillopacityalpha ps:: 0.0 3.83093 moveto ps:: 0.0 253.24313 lineto ps:: 0.99628 253.24313 lineto ps:: 0.99628 3.83093 lineto =ps:: 0.99628 2.26538 2.26538 0.99628 3.83093 0.99628 curveto ps:: 464.17508 0.99628 lineto Cps:: 465.74063 0.99628 467.00974 2.26538 467.00974 3.83093 curveto ps:: 467.00974 253.24313 lineto ps:: 468.00601 253.24313 lineto ps:: 468.00601 3.83093 lineto ;ps:: 468.00601 1.71513 466.29086 0.0 464.17508 0.0 curveto ps:: 3.83093 0.0 lineto 1ps:: 1.71513 0.0 0.0 1.71513 0.0 3.83093 curveto ps:: closepath ,ps:: gsave pgffc pgfeofill grestore newpath ps:: restore ps:: save ps:: /pgffc{0.95 setgray}def !ps:: 0.0 .pgfsetfillopacityalpha ps:: 0.99628 3.83093 moveto ps:: 0.99628 253.24312 lineto ps:: 467.00972 253.24312 lineto ps:: 467.00972 3.83093 lineto Cps:: 467.00972 2.26538 465.74062 0.99628 464.17506 0.99628 curveto ps:: 3.83093 0.99628 lineto =ps:: 2.26538 0.99628 0.99628 2.26538 0.99628 3.83093 curveto ps:: closepath *ps:: gsave pgffc pgffill grestore newpath ps:: restore ps:: save !ps:: 1.0 .pgfsetfillopacityalpha ps:: save 0ps:: [1.0 0.0 0.0 1.0 12.33499 6.66562 ] concat ps:: pgfs ps::[end]color push gray 0goandfconsidertheimagesBd(BrM)B;1(Bs). NotethatBJq=mBJisanBR=m-fmoMdule,andsinceBR=mZES=BkX?,BJq=mBJisinfactaBk-v!ectorspace.SincefBd(BrM)B;1(Bs)generateBJq=mBJasaBkX?-moMdule,dim݁ȮCkDBJq=mBJDE 2B: Butthisisacon!tradiction,sincewecanproMduce3BkX?-linearlyindepMenden!telementsinBJq=mBJ :namelyfBd(Bxz@1Bxz@2)B;1(Bxz@1Bxz@3)B;1(Bxz@2Bxz@3).SuppMosefthereexistB zCi @suc!hthat B z@1Bd(Bxz@1Bxz@2)n+B z@2B(Bxz@1Bxz@3)n+B z@3B(Bxz@2Bxz@3) =0E2BJq=mBJ"E(,)NB z@1Bxz@1Bxz@2.+nB z@2Bxz@1Bxz@3+nB z@3Bxz@2Bxz@3ʧE2 mBJ%;Butfw!ecanthennotethatE^mBJD= EhLBxz@1B;1xz@2B:xz@30;Ei6Ph:Bxz@1Bxz@2B;1xz@1Bxz@3B;xz@2Bxz@3[sEi== D 'Bxz@2:j1Bxz@2B;bxz@2:j1Bxz@3B;xz@1Bxz@2Bxz@3B;1Eups:: 10.69238 5.82552 10.8247 6.08392 10.8247 6.08392 curveto >ps:: 10.5517 6.00443 10.4573 5.80759 10.57538 5.81638 curveto ps:: 8.95343 5.26703 moveto ;ps:: 9.10353 5.32288 9.2618 5.62242 9.2618 5.62242 curveto =ps:: 8.86557 5.46384 8.80365 5.21133 8.95343 5.26703 curveto ps:: 7.32346 4.70905 moveto ;ps:: 7.4614 4.7649 7.55548 5.07307 7.55548 5.07307 curveto ;ps:: 7.2764 4.92348 7.18536 4.6532 7.32346 4.70905 curveto ps:: 5.52835 3.87196 moveto ;ps:: 5.58722 3.81293 5.68996 3.9043 5.68996 3.9043 curveto =ps:: 5.84854 4.05696 5.91335 4.33893 5.91335 4.33893 curveto =ps:: 5.62532 4.18321 5.46962 3.93054 5.52835 3.87196 curveto ps:: 0.28142 0.1518 moveto ps:: 0.72772 0.16045 lineto =ps:: 0.72772 0.16045 1.68256 1.19595 3.03905 1.83842 curveto =ps:: 4.39555 2.48108 5.88565 2.14183 6.09113 2.67725 curveto =ps:: 6.09113 2.67725 5.36833 2.60588 5.32368 2.92737 curveto ps:: 11.38336 5.77429 11.66884 5.97063 12.142 5.75655 curveto =ps:: 12.142 5.75655 12.68623 5.8727 13.14117 6.08665 curveto @ps:: 13.59659 6.30092 13.66507 6.37485 13.61276 6.53201 curveto @ps:: 13.61276 6.53201 12.61421 6.31517 11.21867 5.85126 curveto =ps:: 9.82362 5.38737 9.04927 5.11423 7.49675 4.43332 curveto =ps:: 5.94423 3.75275 3.62906 2.48442 2.49913 1.77747 curveto ;ps:: 1.36955 1.07065 0.28142 0.1518 0.28142 0.1518 curveto ps:: 2.50331 1.87411 moveto ps:: 9.17027 5.28336 10.31232 5.6402 11.65985 6.05081 curveto =ps:: 13.0077 6.46127 13.6054 6.55968 13.6054 6.55968 curveto >ps:: 13.6054 6.55968 13.59659 6.80965 13.09685 6.7562 curveto @ps:: 12.59709 6.70258 11.91876 6.26524 11.91876 6.26524 curveto >ps:: 12.16888 6.48814 12.16888 6.6757 12.16888 6.6757 curveto >ps:: 11.14221 6.40797 10.92809 6.14908 10.77672 5.926 curveto @ps:: 10.62451 5.70308 10.50883 5.73878 10.50883 5.73878 curveto >ps:: 10.39264 5.81892 10.49971 6.03305 10.7852 6.1313 curveto >ps:: 11.07098 6.22923 11.0532 6.57744 11.0532 6.57744 curveto ?ps:: 10.73174 6.26524 10.14285 6.01529 9.76808 5.88135 curveto =ps:: 9.39316 5.74727 9.34853 5.65828 9.34853 5.65828 curveto =ps:: 9.08975 5.11389 8.69691 5.09596 8.83966 5.36353 curveto =ps:: 8.98256 5.63155 9.29506 5.68532 9.29506 5.68532 curveto :ps:: 9.43782 5.8279 9.49126 6.1758 9.49126 6.1758 curveto ps:: 9.50919 5.89928 10.6871 6.38127 11.13356 6.76515 curveto ps:: 11.07082 6.36348 lineto @ps:: 11.07082 6.36348 11.63312 6.66705 12.32924 6.86342 curveto ps:: 12.222 6.55984 lineto ;ps:: 12.222 6.55984 13.31958 7.04181 13.659 6.6757 curveto =ps:: 13.99808 6.30988 12.84673 5.81 12.31097 5.67636 curveto ?ps:: 12.31097 5.67636 12.5254 5.56003 12.20409 5.23009 curveto >ps:: 11.88292 4.89993 11.01704 4.8017 11.01704 4.8017 curveto ?ps:: 11.01704 4.8017 11.24062 4.46262 10.59796 4.18576 curveto ;ps:: 9.9553 3.9091 9.34853 3.93597 9.34853 3.93597 curveto ;ps:: 9.34853 3.93597 9.4023 3.48056 8.7419 3.31989 curveto =ps:: 8.08115 3.15941 7.57262 3.23972 7.57262 3.23972 curveto =ps:: 7.57262 3.23972 7.59038 2.92737 7.16183 2.76686 curveto =ps:: 6.73346 2.60603 6.13562 2.65982 6.13562 2.65982 curveto =ps:: 6.13562 2.65982 6.09113 2.25815 5.05579 2.15094 curveto ps:: gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore ps:: newpath ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]/color push gray 01L color popvgcolor push gray 0WExercises+-color push gray 0X6 color pop -ɍ냍color push gray 0ff color pop-ɍ냍color push gray 0f color popG color pop6@Oᎎ^[ᎎps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if endP ps:SDict begin H.S endps:SDict begin H.R endDps:SDict begin [/View [/XYZ H.V]/Dest (page.7) cvn /DEST pdfmark endɿ1덍color push gray 0iFC냍Tvgcolor push gray 0 =color push gray 0.6color push gray 0.6ٚ$_Ž color popcolor push gray 0LGcolor push gray 12 color popןٚ$ǎ_ color pop_ color pop3Exercises3]T color pop67냍Tcolor push gray 03]T color pop;+8 Tcolor push gray 03]T3]T color popG color pop{ups:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.16) cvn /DEST pdfmark endps:SDict begin H.S endps:SDict begin 13.6 H.A endLps:SDict begin [/View [/XYZ H.V]/Dest (exercise.1.0.8) cvn /DEST pdfmark end color push gray 0jExercise`1.0.8k(ExtrZa):# color popLetnBJHEޯBkX?[Bxz@1B;1E?B;1xzCnP]bMeanideal,:andndacoun!terexampletoBI(BV(BJ )) = 6Ep 6p   [BJmwhenfBkisnotalgebraicallyclosed.,΍z䍍 vps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save #ps:: 0 0.87 0.68 0.32 setcmykcolor ps:: 5.61972 3.10226 moveto ;ps:: 5.38145 3.14342 5.3347 2.9173 5.43457 2.83807 curveto ;ps:: 5.53458 2.75871 5.9751 2.83504 5.9751 2.83504 curveto ps:: 10.69238 5.82552 10.8247 6.08392 10.8247 6.08392 curveto >ps:: 10.5517 6.00443 10.4573 5.80759 10.57538 5.81638 curveto ps:: 8.95343 5.26703 moveto ;ps:: 9.10353 5.32288 9.2618 5.62242 9.2618 5.62242 curveto =ps:: 8.86557 5.46384 8.80365 5.21133 8.95343 5.26703 curveto ps:: 7.32346 4.70905 moveto ;ps:: 7.4614 4.7649 7.55548 5.07307 7.55548 5.07307 curveto ;ps:: 7.2764 4.92348 7.18536 4.6532 7.32346 4.70905 curveto ps:: 5.52835 3.87196 moveto ;ps:: 5.58722 3.81293 5.68996 3.9043 5.68996 3.9043 curveto =ps:: 5.84854 4.05696 5.91335 4.33893 5.91335 4.33893 curveto =ps:: 5.62532 4.18321 5.46962 3.93054 5.52835 3.87196 curveto ps:: 0.28142 0.1518 moveto ps:: 0.72772 0.16045 lineto =ps:: 0.72772 0.16045 1.68256 1.19595 3.03905 1.83842 curveto =ps:: 4.39555 2.48108 5.88565 2.14183 6.09113 2.67725 curveto =ps:: 6.09113 2.67725 5.36833 2.60588 5.32368 2.92737 curveto ps:: 11.38336 5.77429 11.66884 5.97063 12.142 5.75655 curveto =ps:: 12.142 5.75655 12.68623 5.8727 13.14117 6.08665 curveto @ps:: 13.59659 6.30092 13.66507 6.37485 13.61276 6.53201 curveto @ps:: 13.61276 6.53201 12.61421 6.31517 11.21867 5.85126 curveto =ps:: 9.82362 5.38737 9.04927 5.11423 7.49675 4.43332 curveto =ps:: 5.94423 3.75275 3.62906 2.48442 2.49913 1.77747 curveto ;ps:: 1.36955 1.07065 0.28142 0.1518 0.28142 0.1518 curveto ps:: 2.50331 1.87411 moveto ps:: 9.17027 5.28336 10.31232 5.6402 11.65985 6.05081 curveto =ps:: 13.0077 6.46127 13.6054 6.55968 13.6054 6.55968 curveto >ps:: 13.6054 6.55968 13.59659 6.80965 13.09685 6.7562 curveto @ps:: 12.59709 6.70258 11.91876 6.26524 11.91876 6.26524 curveto >ps:: 12.16888 6.48814 12.16888 6.6757 12.16888 6.6757 curveto >ps:: 11.14221 6.40797 10.92809 6.14908 10.77672 5.926 curveto @ps:: 10.62451 5.70308 10.50883 5.73878 10.50883 5.73878 curveto >ps:: 10.39264 5.81892 10.49971 6.03305 10.7852 6.1313 curveto >ps:: 11.07098 6.22923 11.0532 6.57744 11.0532 6.57744 curveto ?ps:: 10.73174 6.26524 10.14285 6.01529 9.76808 5.88135 curveto =ps:: 9.39316 5.74727 9.34853 5.65828 9.34853 5.65828 curveto =ps:: 9.08975 5.11389 8.69691 5.09596 8.83966 5.36353 curveto =ps:: 8.98256 5.63155 9.29506 5.68532 9.29506 5.68532 curveto :ps:: 9.43782 5.8279 9.49126 6.1758 9.49126 6.1758 curveto ps:: 9.50919 5.89928 10.6871 6.38127 11.13356 6.76515 curveto ps:: 11.07082 6.36348 lineto @ps:: 11.07082 6.36348 11.63312 6.66705 12.32924 6.86342 curveto ps:: 12.222 6.55984 lineto ;ps:: 12.222 6.55984 13.31958 7.04181 13.659 6.6757 curveto =ps:: 13.99808 6.30988 12.84673 5.81 12.31097 5.67636 curveto ?ps:: 12.31097 5.67636 12.5254 5.56003 12.20409 5.23009 curveto >ps:: 11.88292 4.89993 11.01704 4.8017 11.01704 4.8017 curveto ?ps:: 11.01704 4.8017 11.24062 4.46262 10.59796 4.18576 curveto ;ps:: 9.9553 3.9091 9.34853 3.93597 9.34853 3.93597 curveto ;ps:: 9.34853 3.93597 9.4023 3.48056 8.7419 3.31989 curveto =ps:: 8.08115 3.15941 7.57262 3.23972 7.57262 3.23972 curveto =ps:: 7.57262 3.23972 7.59038 2.92737 7.16183 2.76686 curveto =ps:: 6.73346 2.60603 6.13562 2.65982 6.13562 2.65982 curveto =ps:: 6.13562 2.65982 6.09113 2.25815 5.05579 2.15094 curveto ps:: gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore ps:: newpath ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]v ps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save ps:: save ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 2ps:: [1.0 0.0 0.0 1.0 234.00299 179.0517 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 2ps:: [1.0 0.0 0.0 1.0 234.00299 179.0517 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: save &ps:: -16322.99998 -16322.99998 moveto %ps:: 16322.99998 -16322.99998 lineto $ps:: 16322.99998 16322.99998 lineto %ps:: -16322.99998 16322.99998 lineto ps:: closepath ps:: 0.0 3.83093 moveto ps:: 0.0 354.27245 lineto =ps:: 0.0 356.38824 1.71513 358.1034 3.83093 358.1034 curveto ps:: 464.17508 358.1034 lineto Hps:: 466.29086 358.1034 468.00601 356.38824 468.00601 354.27245 curveto ps:: 468.00601 3.83093 lineto ;ps:: 468.00601 1.71513 466.29086 0.0 464.17508 0.0 curveto ps:: 3.83093 0.0 lineto 1ps:: 1.71513 0.0 0.0 1.71513 0.0 3.83093 curveto ps:: closepath ps:: clip ps:: newpath ps:: restore ps:: save .ps:: /pgffc{0.62746 0.76863 1 setrgbcolor}def !ps:: 1.0 .pgfsetfillopacityalpha ps:: 0.0 3.83093 moveto ps:: 0.0 354.27245 lineto =ps:: 0.0 356.38824 1.71513 358.1034 3.83093 358.1034 curveto ps:: 464.17508 358.1034 lineto Hps:: 466.29086 358.1034 468.00601 356.38824 468.00601 354.27245 curveto ps:: 468.00601 3.83093 lineto ;ps:: 468.00601 1.71513 466.29086 0.0 464.17508 0.0 curveto ps:: 3.83093 0.0 lineto 1ps:: 1.71513 0.0 0.0 1.71513 0.0 3.83093 curveto ps:: closepath ps:: 0.99628 3.83093 moveto ps:: 0.99628 354.27245 lineto Cps:: 0.99628 355.83801 2.26538 357.10712 3.83093 357.10712 curveto ps:: 464.17508 357.10712 lineto Ips:: 465.74063 357.10712 467.00974 355.83801 467.00974 354.27245 curveto ps:: 467.00974 3.83093 lineto Cps:: 467.00974 2.26538 465.74063 0.99628 464.17508 0.99628 curveto ps:: 3.83093 0.99628 lineto =ps:: 2.26538 0.99628 0.99628 2.26538 0.99628 3.83093 curveto ps:: closepath ,ps:: gsave pgffc pgfeofill grestore newpath ps:: restore ps:: save ps:: /pgffc{0.95 setgray}def !ps:: 0.0 .pgfsetfillopacityalpha ps:: 0.99628 3.83093 moveto ps:: 0.99628 354.27245 lineto Cps:: 0.99628 355.83801 2.26538 357.10712 3.83093 357.10712 curveto ps:: 464.17508 357.10712 lineto Ips:: 465.74063 357.10712 467.00974 355.83801 467.00974 354.27245 curveto ps:: 467.00974 3.83093 lineto Cps:: 467.00974 2.26538 465.74063 0.99628 464.17508 0.99628 curveto ps:: 3.83093 0.99628 lineto =ps:: 2.26538 0.99628 0.99628 2.26538 0.99628 3.83093 curveto ps:: closepath *ps:: gsave pgffc pgffill grestore newpath ps:: restore ps:: save !ps:: 1.0 .pgfsetfillopacityalpha ps:: save 0ps:: [1.0 0.0 0.0 1.0 12.33499 6.66562 ] concat ps:: pgfs ps::[end]color push gray 0~ps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.17) cvn /DEST pdfmark endcolor push gray 0jSolution:  color popTfak!eԙBJeB=WD tBxz@2.+n1'QE2HE?KR[Bx],%notingthatԘBJ6isnontrivialԘandpropMerbutKRisnotalgebraicallyyclosed.ThenfBVn(BJ ) EKRisempt!yf,andthusBI(BVn(BJ )) =BI(E;).ps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.18) cvn /DEST pdfmark ende3color push gray 0jClaim:U color popBI(BVn(BJ )) =KR[Bx].Chec!kingfdenitions,foranysetBXE KAzCnPB=kwehaveTvyxBI(BX) =n SBf8cE2KR[Bx]C.f f f !E8BxE2BX@<;bf-(Bx)=0o.andfsow!evdDacuouslyhaveQíBI(E;) =n SBf8cE2KR[Bx]C.f f f !E8BxE2;B;bf-(Bx)=0ٯo=EfABf8cE2KR[Bx]0{Eg9B=KR[Bx]B:ps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.19) cvn /DEST pdfmark end color push gray 0jClaim:U color pop6Ep 6p   [BJJE6= KR[Bx].Thisyfollo!wsfromthezfactthatmaximalidealsareradical,FandKR[Bx]B=JChECa=yKCbMeingaeldimpliesfthatBJismaximal.Inthiscase6Ep g6p   [BJ= BJDE6=KR[Bx].Thatmaximalidealsareradicalfollo!wsfromthefactthatifBJHEޯBR/ismaximal,aw!ehaveBJDE 6p *6p   [BJUUE BRwhic!hfforces6Ep g6p   [BJ=BJor6Ep g6p   [BJ=BR.Butfif6Ep g6p   [BJ= BR,thenW>1 E26p *6p   [BJ_=,E)N1zCnE2BJforfsome0Bn=,E)1E2BJ"=,E)BJD=BR;con!tradictingftheassumptionthatBJismaximalandthuspropMerbydenition.) color pop ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]Gcolor push gray 0 color popps:SDict begin H.S endps:SDict begin H.R endIps:SDict begin [/View [/XYZ H.V]/Dest (exercises-1) cvn /DEST pdfmark endps:SDict begin H.S endps:SDict begin 13.6 H.A endGps:SDict begin [/View [/XYZ H.V]/Dest (section.2) cvn /DEST pdfmark endcolor push gray 0& 7ȍ h2Mps: currentpoint currentpoint translate 2 2 scale neg exch neg exch translate2Yps: currentpoint currentpoint translate 1 2 div 1 2 div scale neg exch neg exch translate4 7ȄH6CExercises 7 color pop ps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.20) cvn /DEST pdfmark endps:SDict begin H.S endps:SDict begin 13.6 H.A endLps:SDict begin [/View [/XYZ H.V]/Dest (exercise.2.0.1) cvn /DEST pdfmark endcolor push gray 0jExercise22.0.1k(Gathmann2.17):# color popFindftheirreduciblecompMonen!tsofPBX= BVn(BxnEBydz{I;1xz5EByz@2$) EKAz@3B=KCB:,΍z䍍 vps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save #ps:: 0 0.87 0.68 0.32 setcmykcolor ps:: 5.61972 3.10226 moveto ;ps:: 5.38145 3.14342 5.3347 2.9173 5.43457 2.83807 curveto ;ps:: 5.53458 2.75871 5.9751 2.83504 5.9751 2.83504 curveto ps:: 10.69238 5.82552 10.8247 6.08392 10.8247 6.08392 curveto >ps:: 10.5517 6.00443 10.4573 5.80759 10.57538 5.81638 curveto ps:: 8.95343 5.26703 moveto ;ps:: 9.10353 5.32288 9.2618 5.62242 9.2618 5.62242 curveto =ps:: 8.86557 5.46384 8.80365 5.21133 8.95343 5.26703 curveto ps:: 7.32346 4.70905 moveto ;ps:: 7.4614 4.7649 7.55548 5.07307 7.55548 5.07307 curveto ;ps:: 7.2764 4.92348 7.18536 4.6532 7.32346 4.70905 curveto ps:: 5.52835 3.87196 moveto ;ps:: 5.58722 3.81293 5.68996 3.9043 5.68996 3.9043 curveto =ps:: 5.84854 4.05696 5.91335 4.33893 5.91335 4.33893 curveto =ps:: 5.62532 4.18321 5.46962 3.93054 5.52835 3.87196 curveto ps:: 0.28142 0.1518 moveto ps:: 0.72772 0.16045 lineto =ps:: 0.72772 0.16045 1.68256 1.19595 3.03905 1.83842 curveto =ps:: 4.39555 2.48108 5.88565 2.14183 6.09113 2.67725 curveto =ps:: 6.09113 2.67725 5.36833 2.60588 5.32368 2.92737 curveto ps:: 11.38336 5.77429 11.66884 5.97063 12.142 5.75655 curveto =ps:: 12.142 5.75655 12.68623 5.8727 13.14117 6.08665 curveto @ps:: 13.59659 6.30092 13.66507 6.37485 13.61276 6.53201 curveto @ps:: 13.61276 6.53201 12.61421 6.31517 11.21867 5.85126 curveto =ps:: 9.82362 5.38737 9.04927 5.11423 7.49675 4.43332 curveto =ps:: 5.94423 3.75275 3.62906 2.48442 2.49913 1.77747 curveto ;ps:: 1.36955 1.07065 0.28142 0.1518 0.28142 0.1518 curveto ps:: 2.50331 1.87411 moveto ps:: 9.17027 5.28336 10.31232 5.6402 11.65985 6.05081 curveto =ps:: 13.0077 6.46127 13.6054 6.55968 13.6054 6.55968 curveto >ps:: 13.6054 6.55968 13.59659 6.80965 13.09685 6.7562 curveto @ps:: 12.59709 6.70258 11.91876 6.26524 11.91876 6.26524 curveto >ps:: 12.16888 6.48814 12.16888 6.6757 12.16888 6.6757 curveto >ps:: 11.14221 6.40797 10.92809 6.14908 10.77672 5.926 curveto @ps:: 10.62451 5.70308 10.50883 5.73878 10.50883 5.73878 curveto >ps:: 10.39264 5.81892 10.49971 6.03305 10.7852 6.1313 curveto >ps:: 11.07098 6.22923 11.0532 6.57744 11.0532 6.57744 curveto ?ps:: 10.73174 6.26524 10.14285 6.01529 9.76808 5.88135 curveto =ps:: 9.39316 5.74727 9.34853 5.65828 9.34853 5.65828 curveto =ps:: 9.08975 5.11389 8.69691 5.09596 8.83966 5.36353 curveto =ps:: 8.98256 5.63155 9.29506 5.68532 9.29506 5.68532 curveto :ps:: 9.43782 5.8279 9.49126 6.1758 9.49126 6.1758 curveto ps:: 9.50919 5.89928 10.6871 6.38127 11.13356 6.76515 curveto ps:: 11.07082 6.36348 lineto @ps:: 11.07082 6.36348 11.63312 6.66705 12.32924 6.86342 curveto ps:: 12.222 6.55984 lineto ;ps:: 12.222 6.55984 13.31958 7.04181 13.659 6.6757 curveto =ps:: 13.99808 6.30988 12.84673 5.81 12.31097 5.67636 curveto ?ps:: 12.31097 5.67636 12.5254 5.56003 12.20409 5.23009 curveto >ps:: 11.88292 4.89993 11.01704 4.8017 11.01704 4.8017 curveto ?ps:: 11.01704 4.8017 11.24062 4.46262 10.59796 4.18576 curveto ;ps:: 9.9553 3.9091 9.34853 3.93597 9.34853 3.93597 curveto ;ps:: 9.34853 3.93597 9.4023 3.48056 8.7419 3.31989 curveto =ps:: 8.08115 3.15941 7.57262 3.23972 7.57262 3.23972 curveto =ps:: 7.57262 3.23972 7.59038 2.92737 7.16183 2.76686 curveto =ps:: 6.73346 2.60603 6.13562 2.65982 6.13562 2.65982 curveto =ps:: 6.13562 2.65982 6.09113 2.25815 5.05579 2.15094 curveto ps:: gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore ps:: newpath ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]~ps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save ps:: save ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 1ps:: [1.0 0.0 0.0 1.0 234.00299 60.8758 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 1ps:: [1.0 0.0 0.0 1.0 234.00299 60.8758 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: save &ps:: -16322.99998 -16322.99998 moveto %ps:: 16322.99998 -16322.99998 lineto $ps:: 16322.99998 16322.99998 lineto %ps:: -16322.99998 16322.99998 lineto ps:: closepath ps:: 0.0 0.0 moveto ps:: 0.0 117.92065 lineto =ps:: 0.0 120.03645 1.71513 121.7516 3.83093 121.7516 curveto ps:: 464.17508 121.7516 lineto Hps:: 466.29086 121.7516 468.00601 120.03645 468.00601 117.92065 curveto ps:: 468.00601 0.0 lineto ps:: closepath ps:: clip ps:: newpath ps:: restore ps:: save .ps:: /pgffc{0.62746 0.76863 1 setrgbcolor}def !ps:: 1.0 .pgfsetfillopacityalpha ps:: 0.0 0.0 moveto ps:: 0.0 117.92065 lineto =ps:: 0.0 120.03645 1.71513 121.7516 3.83093 121.7516 curveto ps:: 464.17508 121.7516 lineto Hps:: 466.29086 121.7516 468.00601 120.03645 468.00601 117.92065 curveto ps:: 468.00601 0.0 lineto ps:: 467.00974 0.0 lineto ps:: 467.00974 117.92065 lineto Ips:: 467.00974 119.48622 465.74063 120.75533 464.17508 120.75533 curveto ps:: 3.83093 120.75533 lineto Cps:: 2.26538 120.75533 0.99628 119.48622 0.99628 117.92065 curveto ps:: 0.99628 0.0 lineto ps:: closepath ,ps:: gsave pgffc pgfeofill grestore newpath ps:: restore ps:: save ps:: /pgffc{0.95 setgray}def !ps:: 0.0 .pgfsetfillopacityalpha ps:: 0.99628 0.0 moveto ps:: 0.99628 117.92065 lineto Cps:: 0.99628 119.48622 2.26538 120.75533 3.83093 120.75533 curveto ps:: 464.17506 120.75533 lineto Ips:: 465.74062 120.75533 467.00972 119.48622 467.00972 117.92065 curveto ps:: 467.00972 0.0 lineto ps:: closepath *ps:: gsave pgffc pgffill grestore newpath ps:: restore ps:: save !ps:: 1.0 .pgfsetfillopacityalpha ps:: save 0ps:: [1.0 0.0 0.0 1.0 12.33499 6.66562 ] concat ps:: pgfs ps::[end]color push gray 0,ps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.21) cvn /DEST pdfmark endcolor push gray 0jSolution: color popSincefBx =Bydz!forallpMoin!tsinBX,wehave]YBXR= BVn(BxnEBydz{I;1yzz@25EnByz@2$)NR= BVB  ;BxnEBydz{I;1y(Bzz@25EnBy)Y'tR= BVn(BxnEBydz{I;1y)nE[BV(BxEBydz{I;1zz@25EnBy)R𞞩:= BXz@1.E[nBXz@2B: ps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.22) cvn /DEST pdfmark end) color pop ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]/color push gray 01L color popvgcolor push gray 0WExercises+-color push gray 0X7 color pop -ɍ냍color push gray 0ff color pop-ɍ냍color push gray 0f color popG color pop6@Oᎎ󍍟ᎎps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if endP ps:SDict begin H.S endps:SDict begin H.R endDps:SDict begin [/View [/XYZ H.V]/Dest (page.8) cvn /DEST pdfmark endɿ1덍color push gray 0iFC냍Tvgcolor push gray 0 =color push gray 0.6color push gray 0.6ٚ$_Ž color popcolor push gray 0LGcolor push gray 12 color popןٚ$ǎ_ color pop_ color pop3Exercises3]T color pop67냍Tcolor push gray 03]T color pop;+8 Tcolor push gray 03]T3]T color popG color pop{bps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save ps:: save ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 2ps:: [1.0 0.0 0.0 1.0 234.00299 69.64711 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 2ps:: [1.0 0.0 0.0 1.0 234.00299 69.64711 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: save &ps:: -16322.99998 -16322.99998 moveto %ps:: 16322.99998 -16322.99998 lineto $ps:: 16322.99998 16322.99998 lineto %ps:: -16322.99998 16322.99998 lineto ps:: closepath ps:: 0.0 3.83093 moveto ps:: 0.0 139.29425 lineto ps:: 468.00601 139.29425 lineto ps:: 468.00601 3.83093 lineto ;ps:: 468.00601 1.71513 466.29086 0.0 464.17508 0.0 curveto ps:: 3.83093 0.0 lineto 1ps:: 1.71513 0.0 0.0 1.71513 0.0 3.83093 curveto ps:: closepath ps:: clip ps:: newpath ps:: restore ps:: save .ps:: /pgffc{0.62746 0.76863 1 setrgbcolor}def !ps:: 1.0 .pgfsetfillopacityalpha ps:: 0.0 3.83093 moveto ps:: 0.0 139.29425 lineto ps:: 0.99628 139.29425 lineto ps:: 0.99628 3.83093 lineto =ps:: 0.99628 2.26538 2.26538 0.99628 3.83093 0.99628 curveto ps:: 464.17508 0.99628 lineto Cps:: 465.74063 0.99628 467.00974 2.26538 467.00974 3.83093 curveto ps:: 467.00974 139.29425 lineto ps:: 468.00601 139.29425 lineto ps:: 468.00601 3.83093 lineto ;ps:: 468.00601 1.71513 466.29086 0.0 464.17508 0.0 curveto ps:: 3.83093 0.0 lineto 1ps:: 1.71513 0.0 0.0 1.71513 0.0 3.83093 curveto ps:: closepath ,ps:: gsave pgffc pgfeofill grestore newpath ps:: restore ps:: save ps:: /pgffc{0.95 setgray}def !ps:: 0.0 .pgfsetfillopacityalpha ps:: 0.99628 3.83093 moveto ps:: 0.99628 139.29424 lineto ps:: 467.00972 139.29424 lineto ps:: 467.00972 3.83093 lineto Cps:: 467.00972 2.26538 465.74062 0.99628 464.17506 0.99628 curveto ps:: 3.83093 0.99628 lineto =ps:: 2.26538 0.99628 0.99628 2.26538 0.99628 3.83093 curveto ps:: closepath *ps:: gsave pgffc pgffill grestore newpath ps:: restore ps:: save !ps:: 1.0 .pgfsetfillopacityalpha ps:: save 0ps:: [1.0 0.0 0.0 1.0 12.33499 8.78683 ] concat ps:: pgfs ps::[end]color push gray 0Ycolor push gray 0jClaim:U color popTheseft!wosubvdDarietiesareirreducible. Itfsucestosho!wthattheBA(BXzCid)areintegraldomains.WfehaveaBA(BXz@1) :Zp= KC[Bx;1yd;z{I]B=EhTBxnEByz{I;1y/.Ei66=BKC[By;z{I]B=EhTByEi%% =KC[Bz{I]B;whic!hfisanintegraldomainsinceKCisaeldandthusanintegraldomain,andFMBA(BXz@2) :Zp= KC[Bx;1yd;z{I]B=DBxnEByz{I;1zz@25EnByI?ERfERf=]KC[By;z{I]B=DBzz@25EnBy%gE.E.=:KC[Byd]B;Gwhic!hfisanintegraldomainforthesamereason.) color pop ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]color push gray 0 color popps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.23) cvn /DEST pdfmark endps:SDict begin H.S endps:SDict begin 13.6 H.A endLps:SDict begin [/View [/XYZ H.V]/Dest (exercise.2.0.2) cvn /DEST pdfmark end34color push gray 0jExercise22.0.2k(Gathmann2.18):# color popLetfBXE KAzCn NbMeanarbitrarysubsetandsho!wthat\BVn(BI(BX)) =?QMfe(hBX :"Ǎz䍍 vps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save #ps:: 0 0.87 0.68 0.32 setcmykcolor ps:: 5.61972 3.10226 moveto ;ps:: 5.38145 3.14342 5.3347 2.9173 5.43457 2.83807 curveto ;ps:: 5.53458 2.75871 5.9751 2.83504 5.9751 2.83504 curveto ps:: 10.69238 5.82552 10.8247 6.08392 10.8247 6.08392 curveto >ps:: 10.5517 6.00443 10.4573 5.80759 10.57538 5.81638 curveto ps:: 8.95343 5.26703 moveto ;ps:: 9.10353 5.32288 9.2618 5.62242 9.2618 5.62242 curveto =ps:: 8.86557 5.46384 8.80365 5.21133 8.95343 5.26703 curveto ps:: 7.32346 4.70905 moveto ;ps:: 7.4614 4.7649 7.55548 5.07307 7.55548 5.07307 curveto ;ps:: 7.2764 4.92348 7.18536 4.6532 7.32346 4.70905 curveto ps:: 5.52835 3.87196 moveto ;ps:: 5.58722 3.81293 5.68996 3.9043 5.68996 3.9043 curveto =ps:: 5.84854 4.05696 5.91335 4.33893 5.91335 4.33893 curveto =ps:: 5.62532 4.18321 5.46962 3.93054 5.52835 3.87196 curveto ps:: 0.28142 0.1518 moveto ps:: 0.72772 0.16045 lineto =ps:: 0.72772 0.16045 1.68256 1.19595 3.03905 1.83842 curveto =ps:: 4.39555 2.48108 5.88565 2.14183 6.09113 2.67725 curveto =ps:: 6.09113 2.67725 5.36833 2.60588 5.32368 2.92737 curveto ps:: 11.38336 5.77429 11.66884 5.97063 12.142 5.75655 curveto =ps:: 12.142 5.75655 12.68623 5.8727 13.14117 6.08665 curveto @ps:: 13.59659 6.30092 13.66507 6.37485 13.61276 6.53201 curveto @ps:: 13.61276 6.53201 12.61421 6.31517 11.21867 5.85126 curveto =ps:: 9.82362 5.38737 9.04927 5.11423 7.49675 4.43332 curveto =ps:: 5.94423 3.75275 3.62906 2.48442 2.49913 1.77747 curveto ;ps:: 1.36955 1.07065 0.28142 0.1518 0.28142 0.1518 curveto ps:: 2.50331 1.87411 moveto ps:: 9.17027 5.28336 10.31232 5.6402 11.65985 6.05081 curveto =ps:: 13.0077 6.46127 13.6054 6.55968 13.6054 6.55968 curveto >ps:: 13.6054 6.55968 13.59659 6.80965 13.09685 6.7562 curveto @ps:: 12.59709 6.70258 11.91876 6.26524 11.91876 6.26524 curveto >ps:: 12.16888 6.48814 12.16888 6.6757 12.16888 6.6757 curveto >ps:: 11.14221 6.40797 10.92809 6.14908 10.77672 5.926 curveto @ps:: 10.62451 5.70308 10.50883 5.73878 10.50883 5.73878 curveto >ps:: 10.39264 5.81892 10.49971 6.03305 10.7852 6.1313 curveto >ps:: 11.07098 6.22923 11.0532 6.57744 11.0532 6.57744 curveto ?ps:: 10.73174 6.26524 10.14285 6.01529 9.76808 5.88135 curveto =ps:: 9.39316 5.74727 9.34853 5.65828 9.34853 5.65828 curveto =ps:: 9.08975 5.11389 8.69691 5.09596 8.83966 5.36353 curveto =ps:: 8.98256 5.63155 9.29506 5.68532 9.29506 5.68532 curveto :ps:: 9.43782 5.8279 9.49126 6.1758 9.49126 6.1758 curveto ps:: 9.50919 5.89928 10.6871 6.38127 11.13356 6.76515 curveto ps:: 11.07082 6.36348 lineto @ps:: 11.07082 6.36348 11.63312 6.66705 12.32924 6.86342 curveto ps:: 12.222 6.55984 lineto ;ps:: 12.222 6.55984 13.31958 7.04181 13.659 6.6757 curveto =ps:: 13.99808 6.30988 12.84673 5.81 12.31097 5.67636 curveto ?ps:: 12.31097 5.67636 12.5254 5.56003 12.20409 5.23009 curveto >ps:: 11.88292 4.89993 11.01704 4.8017 11.01704 4.8017 curveto ?ps:: 11.01704 4.8017 11.24062 4.46262 10.59796 4.18576 curveto ;ps:: 9.9553 3.9091 9.34853 3.93597 9.34853 3.93597 curveto ;ps:: 9.34853 3.93597 9.4023 3.48056 8.7419 3.31989 curveto =ps:: 8.08115 3.15941 7.57262 3.23972 7.57262 3.23972 curveto =ps:: 7.57262 3.23972 7.59038 2.92737 7.16183 2.76686 curveto =ps:: 6.73346 2.60603 6.13562 2.65982 6.13562 2.65982 curveto =ps:: 6.13562 2.65982 6.09113 2.25815 5.05579 2.15094 curveto ps:: gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore ps:: newpath ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]*5ps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save ps:: save ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 3ps:: [1.0 0.0 0.0 1.0 234.00299 128.30295 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 3ps:: [1.0 0.0 0.0 1.0 234.00299 128.30295 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: save &ps:: -16322.99998 -16322.99998 moveto %ps:: 16322.99998 -16322.99998 lineto $ps:: 16322.99998 16322.99998 lineto %ps:: -16322.99998 16322.99998 lineto ps:: closepath ps:: 0.0 3.83093 moveto ps:: 0.0 252.77496 lineto ?ps:: 0.0 254.89076 1.71513 256.60591 3.83093 256.60591 curveto ps:: 464.17508 256.60591 lineto Ips:: 466.29086 256.60591 468.00601 254.89076 468.00601 252.77496 curveto ps:: 468.00601 3.83093 lineto ;ps:: 468.00601 1.71513 466.29086 0.0 464.17508 0.0 curveto ps:: 3.83093 0.0 lineto 1ps:: 1.71513 0.0 0.0 1.71513 0.0 3.83093 curveto ps:: closepath ps:: clip ps:: newpath ps:: restore ps:: save .ps:: /pgffc{0.62746 0.76863 1 setrgbcolor}def !ps:: 1.0 .pgfsetfillopacityalpha ps:: 0.0 3.83093 moveto ps:: 0.0 252.77496 lineto ?ps:: 0.0 254.89076 1.71513 256.60591 3.83093 256.60591 curveto ps:: 464.17508 256.60591 lineto Ips:: 466.29086 256.60591 468.00601 254.89076 468.00601 252.77496 curveto ps:: 468.00601 3.83093 lineto ;ps:: 468.00601 1.71513 466.29086 0.0 464.17508 0.0 curveto ps:: 3.83093 0.0 lineto 1ps:: 1.71513 0.0 0.0 1.71513 0.0 3.83093 curveto ps:: closepath ps:: 0.99628 3.83093 moveto ps:: 0.99628 252.77496 lineto Cps:: 0.99628 254.34052 2.26538 255.60963 3.83093 255.60963 curveto ps:: 464.17508 255.60963 lineto Ips:: 465.74063 255.60963 467.00974 254.34052 467.00974 252.77496 curveto ps:: 467.00974 3.83093 lineto Cps:: 467.00974 2.26538 465.74063 0.99628 464.17508 0.99628 curveto ps:: 3.83093 0.99628 lineto =ps:: 2.26538 0.99628 0.99628 2.26538 0.99628 3.83093 curveto ps:: closepath ,ps:: gsave pgffc pgfeofill grestore newpath ps:: restore ps:: save ps:: /pgffc{0.95 setgray}def !ps:: 0.0 .pgfsetfillopacityalpha ps:: 0.99628 3.83093 moveto ps:: 0.99628 252.77496 lineto Cps:: 0.99628 254.34052 2.26538 255.60963 3.83093 255.60963 curveto ps:: 464.17508 255.60963 lineto Ips:: 465.74063 255.60963 467.00974 254.34052 467.00974 252.77496 curveto ps:: 467.00974 3.83093 lineto Cps:: 467.00974 2.26538 465.74063 0.99628 464.17508 0.99628 curveto ps:: 3.83093 0.99628 lineto =ps:: 2.26538 0.99628 0.99628 2.26538 0.99628 3.83093 curveto ps:: closepath *ps:: gsave pgffc pgffill grestore newpath ps:: restore ps:: save !ps:: 1.0 .pgfsetfillopacityalpha ps:: save 0ps:: [1.0 0.0 0.0 1.0 12.33499 8.78683 ] concat ps:: pgfs ps::[end]color push gray 0 ps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.24) cvn /DEST pdfmark endcolor push gray 0jSolution: color pop队QMfe(hBX E BVn(BI(BX)):q Wfeha!veBXEBVn(BI(BX))andsinceBVn(BJ )isclosedintheZariskitopMologyforan!yidealBJHEޯBkX?[Bxz@1B;1E?B;1xzCnP]fb!ydenition,BVn(BI(BX))fisclosed.ThusXBXE BVn(BI(BX))fandBV(BI(BX))fclosed* (=,E)QMfe(hBXME BV(BI(BX))B;sinceQMfe(hBX;$isfthein!tersectionofallclosedsetscontainingBX. BVn(BI(BX)) E?QMfe(hBX :NotingthatBVn(1E )B;1I(E)areindividuallyorder-rev!ersing,cJwendthatBVn(BI(1E ))isorder-prpeservingBandfth!usBXE?QMfe(hBXM=,E)NBVn(BI(BX)) EBV(BI(队QMfe(hBX X))=?QMfe(hBX ;whereinthelastequalit!ywe'veusedpart(i)oftheNullstellensatz:k7ifBX isananevdDariet!yf, thenBVn(BI(BX))S=SBX.`@ThisappliesherebMecauseuQMfe(hBXisalw!aysclosed,ܹandtheclosedsetsintheZariskiftopMologyarepreciselytheanevdDarieties.) color pop ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]Gcolor push gray 0 color popps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.25) cvn /DEST pdfmark endps:SDict begin H.S endps:SDict begin 13.6 H.A endLps:SDict begin [/View [/XYZ H.V]/Dest (exercise.2.0.3) cvn /DEST pdfmark end34color push gray 0jExercisen2.0.3k(GathmannS2.21):# color popLet.EfuBUzCiTEg2ps:: 10.69238 5.82552 10.8247 6.08392 10.8247 6.08392 curveto >ps:: 10.5517 6.00443 10.4573 5.80759 10.57538 5.81638 curveto ps:: 8.95343 5.26703 moveto ;ps:: 9.10353 5.32288 9.2618 5.62242 9.2618 5.62242 curveto =ps:: 8.86557 5.46384 8.80365 5.21133 8.95343 5.26703 curveto ps:: 7.32346 4.70905 moveto ;ps:: 7.4614 4.7649 7.55548 5.07307 7.55548 5.07307 curveto ;ps:: 7.2764 4.92348 7.18536 4.6532 7.32346 4.70905 curveto ps:: 5.52835 3.87196 moveto ;ps:: 5.58722 3.81293 5.68996 3.9043 5.68996 3.9043 curveto =ps:: 5.84854 4.05696 5.91335 4.33893 5.91335 4.33893 curveto =ps:: 5.62532 4.18321 5.46962 3.93054 5.52835 3.87196 curveto ps:: 0.28142 0.1518 moveto ps:: 0.72772 0.16045 lineto =ps:: 0.72772 0.16045 1.68256 1.19595 3.03905 1.83842 curveto =ps:: 4.39555 2.48108 5.88565 2.14183 6.09113 2.67725 curveto =ps:: 6.09113 2.67725 5.36833 2.60588 5.32368 2.92737 curveto ps:: 11.38336 5.77429 11.66884 5.97063 12.142 5.75655 curveto =ps:: 12.142 5.75655 12.68623 5.8727 13.14117 6.08665 curveto @ps:: 13.59659 6.30092 13.66507 6.37485 13.61276 6.53201 curveto @ps:: 13.61276 6.53201 12.61421 6.31517 11.21867 5.85126 curveto =ps:: 9.82362 5.38737 9.04927 5.11423 7.49675 4.43332 curveto =ps:: 5.94423 3.75275 3.62906 2.48442 2.49913 1.77747 curveto ;ps:: 1.36955 1.07065 0.28142 0.1518 0.28142 0.1518 curveto ps:: 2.50331 1.87411 moveto ps:: 9.17027 5.28336 10.31232 5.6402 11.65985 6.05081 curveto =ps:: 13.0077 6.46127 13.6054 6.55968 13.6054 6.55968 curveto >ps:: 13.6054 6.55968 13.59659 6.80965 13.09685 6.7562 curveto @ps:: 12.59709 6.70258 11.91876 6.26524 11.91876 6.26524 curveto >ps:: 12.16888 6.48814 12.16888 6.6757 12.16888 6.6757 curveto >ps:: 11.14221 6.40797 10.92809 6.14908 10.77672 5.926 curveto @ps:: 10.62451 5.70308 10.50883 5.73878 10.50883 5.73878 curveto >ps:: 10.39264 5.81892 10.49971 6.03305 10.7852 6.1313 curveto >ps:: 11.07098 6.22923 11.0532 6.57744 11.0532 6.57744 curveto ?ps:: 10.73174 6.26524 10.14285 6.01529 9.76808 5.88135 curveto =ps:: 9.39316 5.74727 9.34853 5.65828 9.34853 5.65828 curveto =ps:: 9.08975 5.11389 8.69691 5.09596 8.83966 5.36353 curveto =ps:: 8.98256 5.63155 9.29506 5.68532 9.29506 5.68532 curveto :ps:: 9.43782 5.8279 9.49126 6.1758 9.49126 6.1758 curveto ps:: 9.50919 5.89928 10.6871 6.38127 11.13356 6.76515 curveto ps:: 11.07082 6.36348 lineto @ps:: 11.07082 6.36348 11.63312 6.66705 12.32924 6.86342 curveto ps:: 12.222 6.55984 lineto ;ps:: 12.222 6.55984 13.31958 7.04181 13.659 6.6757 curveto =ps:: 13.99808 6.30988 12.84673 5.81 12.31097 5.67636 curveto ?ps:: 12.31097 5.67636 12.5254 5.56003 12.20409 5.23009 curveto >ps:: 11.88292 4.89993 11.01704 4.8017 11.01704 4.8017 curveto ?ps:: 11.01704 4.8017 11.24062 4.46262 10.59796 4.18576 curveto ;ps:: 9.9553 3.9091 9.34853 3.93597 9.34853 3.93597 curveto ;ps:: 9.34853 3.93597 9.4023 3.48056 8.7419 3.31989 curveto =ps:: 8.08115 3.15941 7.57262 3.23972 7.57262 3.23972 curveto =ps:: 7.57262 3.23972 7.59038 2.92737 7.16183 2.76686 curveto =ps:: 6.73346 2.60603 6.13562 2.65982 6.13562 2.65982 curveto =ps:: 6.13562 2.65982 6.09113 2.25815 5.05579 2.15094 curveto ps:: gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore ps:: newpath ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]P1ps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save ps:: save ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 2ps:: [1.0 0.0 0.0 1.0 234.00299 32.67369 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 2ps:: [1.0 0.0 0.0 1.0 234.00299 32.67369 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: save &ps:: -16322.99998 -16322.99998 moveto %ps:: 16322.99998 -16322.99998 lineto $ps:: 16322.99998 16322.99998 lineto %ps:: -16322.99998 16322.99998 lineto ps:: closepath ps:: 0.0 0.0 moveto ps:: 0.0 61.51646 lineto ps:: 0.0 292.6563 1.71513 294.37143 3.83093 294.37143 curveto ps:: 464.17508 294.37143 lineto Gps:: 466.29086 294.37143 468.00601 292.6563 468.00601 290.5405 curveto ps:: 468.00601 3.83093 lineto ;ps:: 468.00601 1.71513 466.29086 0.0 464.17508 0.0 curveto ps:: 3.83093 0.0 lineto 1ps:: 1.71513 0.0 0.0 1.71513 0.0 3.83093 curveto ps:: closepath ps:: clip ps:: newpath ps:: restore ps:: save .ps:: /pgffc{0.62746 0.76863 1 setrgbcolor}def !ps:: 1.0 .pgfsetfillopacityalpha ps:: 0.0 3.83093 moveto ps:: 0.0 290.5405 lineto >ps:: 0.0 292.6563 1.71513 294.37143 3.83093 294.37143 curveto ps:: 464.17508 294.37143 lineto Gps:: 466.29086 294.37143 468.00601 292.6563 468.00601 290.5405 curveto ps:: 468.00601 3.83093 lineto ;ps:: 468.00601 1.71513 466.29086 0.0 464.17508 0.0 curveto ps:: 3.83093 0.0 lineto 1ps:: 1.71513 0.0 0.0 1.71513 0.0 3.83093 curveto ps:: closepath ps:: 0.99628 3.83093 moveto ps:: 0.99628 290.5405 lineto Cps:: 0.99628 292.10605 2.26538 293.37515 3.83093 293.37515 curveto ps:: 464.17508 293.37515 lineto Hps:: 465.74063 293.37515 467.00974 292.10605 467.00974 290.5405 curveto ps:: 467.00974 3.83093 lineto Cps:: 467.00974 2.26538 465.74063 0.99628 464.17508 0.99628 curveto ps:: 3.83093 0.99628 lineto =ps:: 2.26538 0.99628 0.99628 2.26538 0.99628 3.83093 curveto ps:: closepath ,ps:: gsave pgffc pgfeofill grestore newpath ps:: restore ps:: save ps:: /pgffc{0.95 setgray}def !ps:: 0.0 .pgfsetfillopacityalpha ps:: 0.99628 3.83093 moveto ps:: 0.99628 290.5405 lineto Cps:: 0.99628 292.10605 2.26538 293.37515 3.83093 293.37515 curveto ps:: 464.17508 293.37515 lineto Hps:: 465.74063 293.37515 467.00974 292.10605 467.00974 290.5405 curveto ps:: 467.00974 3.83093 lineto Cps:: 467.00974 2.26538 465.74063 0.99628 464.17508 0.99628 curveto ps:: 3.83093 0.99628 lineto =ps:: 2.26538 0.99628 0.99628 2.26538 0.99628 3.83093 curveto ps:: closepath *ps:: gsave pgffc pgffill grestore newpath ps:: restore ps:: save !ps:: 1.0 .pgfsetfillopacityalpha ps:: save 0ps:: [1.0 0.0 0.0 1.0 12.33499 8.78683 ] concat ps:: pgfs ps::[end]color push gray 0 ps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.27) cvn /DEST pdfmark endps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.28) cvn /DEST pdfmark end6fcolor push gray 0jSolutionk(b): color pop!color push gray 0jClaim:U color popBX^isfirreducible E(,)an!ytwoopMensubsetsintersect.Thisfollo!wsbMecauseotherwise,ifBU';1VdxEBXareopenanddisjoin!tthenBX?En3GBU';bX>EnBV;arepropMer and closed.Butthenw!ecanwriteBX*=N ( 4BXJEnnBU&),C0a;1=(?sgBXJEnnBV\S)eχasaunionofpropMerclosedfsubsets,forcingBX^tonotbMeirreducible.Sonitnsucestosho!wthatifBU';1VyE BXIthenBU/E\,BVisnonempt!yf.SinceEfBUzCipEgJH BX,yLwencanndafpairBi;1jFsuc!hthatthereisatleastonepMointinBUE\nBUzCi @andonepMointinBVE\nBUzCjf .Butb!yassumptionBUzCi|E\BUzCj`isnonemptyf,>sobMothBUIE\BUzCi_andBUzCj~E\BUzCi_areopennonempt!ysubsetsoffBUzCid.SinceBUzCi @w!asassumedirreducible,theymustintersect,sothereexistsapMointliBxz@0ʧE2 (LBUE\nBUzCi'ܝ)-E\n(BUzCjE\nBUzCi))0]= BUE\n(BUzCiE\nBUzCj)):37&=;f~ĚBU <:Uq Wfeicanno!wsimilarlyhnotethat;ffE~ĚBUKE\ BVXandBUzCjE\BVXarenonempt!yhopMensubsetsofBVn,5andthusin!tersect.SofthereisapMointX~Xps:: 10.69238 5.82552 10.8247 6.08392 10.8247 6.08392 curveto >ps:: 10.5517 6.00443 10.4573 5.80759 10.57538 5.81638 curveto ps:: 8.95343 5.26703 moveto ;ps:: 9.10353 5.32288 9.2618 5.62242 9.2618 5.62242 curveto =ps:: 8.86557 5.46384 8.80365 5.21133 8.95343 5.26703 curveto ps:: 7.32346 4.70905 moveto ;ps:: 7.4614 4.7649 7.55548 5.07307 7.55548 5.07307 curveto ;ps:: 7.2764 4.92348 7.18536 4.6532 7.32346 4.70905 curveto ps:: 5.52835 3.87196 moveto ;ps:: 5.58722 3.81293 5.68996 3.9043 5.68996 3.9043 curveto =ps:: 5.84854 4.05696 5.91335 4.33893 5.91335 4.33893 curveto =ps:: 5.62532 4.18321 5.46962 3.93054 5.52835 3.87196 curveto ps:: 0.28142 0.1518 moveto ps:: 0.72772 0.16045 lineto =ps:: 0.72772 0.16045 1.68256 1.19595 3.03905 1.83842 curveto =ps:: 4.39555 2.48108 5.88565 2.14183 6.09113 2.67725 curveto =ps:: 6.09113 2.67725 5.36833 2.60588 5.32368 2.92737 curveto ps:: 11.38336 5.77429 11.66884 5.97063 12.142 5.75655 curveto =ps:: 12.142 5.75655 12.68623 5.8727 13.14117 6.08665 curveto @ps:: 13.59659 6.30092 13.66507 6.37485 13.61276 6.53201 curveto @ps:: 13.61276 6.53201 12.61421 6.31517 11.21867 5.85126 curveto =ps:: 9.82362 5.38737 9.04927 5.11423 7.49675 4.43332 curveto =ps:: 5.94423 3.75275 3.62906 2.48442 2.49913 1.77747 curveto ;ps:: 1.36955 1.07065 0.28142 0.1518 0.28142 0.1518 curveto ps:: 2.50331 1.87411 moveto ps:: 9.17027 5.28336 10.31232 5.6402 11.65985 6.05081 curveto =ps:: 13.0077 6.46127 13.6054 6.55968 13.6054 6.55968 curveto >ps:: 13.6054 6.55968 13.59659 6.80965 13.09685 6.7562 curveto @ps:: 12.59709 6.70258 11.91876 6.26524 11.91876 6.26524 curveto >ps:: 12.16888 6.48814 12.16888 6.6757 12.16888 6.6757 curveto >ps:: 11.14221 6.40797 10.92809 6.14908 10.77672 5.926 curveto @ps:: 10.62451 5.70308 10.50883 5.73878 10.50883 5.73878 curveto >ps:: 10.39264 5.81892 10.49971 6.03305 10.7852 6.1313 curveto >ps:: 11.07098 6.22923 11.0532 6.57744 11.0532 6.57744 curveto ?ps:: 10.73174 6.26524 10.14285 6.01529 9.76808 5.88135 curveto =ps:: 9.39316 5.74727 9.34853 5.65828 9.34853 5.65828 curveto =ps:: 9.08975 5.11389 8.69691 5.09596 8.83966 5.36353 curveto =ps:: 8.98256 5.63155 9.29506 5.68532 9.29506 5.68532 curveto :ps:: 9.43782 5.8279 9.49126 6.1758 9.49126 6.1758 curveto ps:: 9.50919 5.89928 10.6871 6.38127 11.13356 6.76515 curveto ps:: 11.07082 6.36348 lineto @ps:: 11.07082 6.36348 11.63312 6.66705 12.32924 6.86342 curveto ps:: 12.222 6.55984 lineto ;ps:: 12.222 6.55984 13.31958 7.04181 13.659 6.6757 curveto =ps:: 13.99808 6.30988 12.84673 5.81 12.31097 5.67636 curveto ?ps:: 12.31097 5.67636 12.5254 5.56003 12.20409 5.23009 curveto >ps:: 11.88292 4.89993 11.01704 4.8017 11.01704 4.8017 curveto ?ps:: 11.01704 4.8017 11.24062 4.46262 10.59796 4.18576 curveto ;ps:: 9.9553 3.9091 9.34853 3.93597 9.34853 3.93597 curveto ;ps:: 9.34853 3.93597 9.4023 3.48056 8.7419 3.31989 curveto =ps:: 8.08115 3.15941 7.57262 3.23972 7.57262 3.23972 curveto =ps:: 7.57262 3.23972 7.59038 2.92737 7.16183 2.76686 curveto =ps:: 6.73346 2.60603 6.13562 2.65982 6.13562 2.65982 curveto =ps:: 6.13562 2.65982 6.09113 2.25815 5.05579 2.15094 curveto ps:: gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore ps:: newpath ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]tps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save ps:: save ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 0ps:: [1.0 0.0 0.0 1.0 234.00299 82.428 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 0ps:: [1.0 0.0 0.0 1.0 234.00299 82.428 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: save &ps:: -16322.99998 -16322.99998 moveto %ps:: 16322.99998 -16322.99998 lineto $ps:: 16322.99998 16322.99998 lineto %ps:: -16322.99998 16322.99998 lineto ps:: closepath ps:: 0.0 3.83093 moveto ps:: 0.0 161.02505 lineto ?ps:: 0.0 163.14084 1.71513 164.85599 3.83093 164.85599 curveto ps:: 464.17508 164.85599 lineto Ips:: 466.29086 164.85599 468.00601 163.14084 468.00601 161.02505 curveto ps:: 468.00601 3.83093 lineto ;ps:: 468.00601 1.71513 466.29086 0.0 464.17508 0.0 curveto ps:: 3.83093 0.0 lineto 1ps:: 1.71513 0.0 0.0 1.71513 0.0 3.83093 curveto ps:: closepath ps:: clip ps:: newpath ps:: restore ps:: save .ps:: /pgffc{0.62746 0.76863 1 setrgbcolor}def !ps:: 1.0 .pgfsetfillopacityalpha ps:: 0.0 3.83093 moveto ps:: 0.0 161.02505 lineto ?ps:: 0.0 163.14084 1.71513 164.85599 3.83093 164.85599 curveto ps:: 464.17508 164.85599 lineto Ips:: 466.29086 164.85599 468.00601 163.14084 468.00601 161.02505 curveto ps:: 468.00601 3.83093 lineto ;ps:: 468.00601 1.71513 466.29086 0.0 464.17508 0.0 curveto ps:: 3.83093 0.0 lineto 1ps:: 1.71513 0.0 0.0 1.71513 0.0 3.83093 curveto ps:: closepath ps:: 0.99628 3.83093 moveto ps:: 0.99628 161.02505 lineto Bps:: 0.99628 162.5906 2.26538 163.85971 3.83093 163.85971 curveto ps:: 464.17508 163.85971 lineto Hps:: 465.74063 163.85971 467.00974 162.5906 467.00974 161.02505 curveto ps:: 467.00974 3.83093 lineto Cps:: 467.00974 2.26538 465.74063 0.99628 464.17508 0.99628 curveto ps:: 3.83093 0.99628 lineto =ps:: 2.26538 0.99628 0.99628 2.26538 0.99628 3.83093 curveto ps:: closepath ,ps:: gsave pgffc pgfeofill grestore newpath ps:: restore ps:: save ps:: /pgffc{0.95 setgray}def !ps:: 0.0 .pgfsetfillopacityalpha ps:: 0.99628 3.83093 moveto ps:: 0.99628 161.02505 lineto Bps:: 0.99628 162.5906 2.26538 163.85971 3.83093 163.85971 curveto ps:: 464.17508 163.85971 lineto Hps:: 465.74063 163.85971 467.00974 162.5906 467.00974 161.02505 curveto ps:: 467.00974 3.83093 lineto Cps:: 467.00974 2.26538 465.74063 0.99628 464.17508 0.99628 curveto ps:: 3.83093 0.99628 lineto =ps:: 2.26538 0.99628 0.99628 2.26538 0.99628 3.83093 curveto ps:: closepath *ps:: gsave pgffc pgffill grestore newpath ps:: restore ps:: save !ps:: 1.0 .pgfsetfillopacityalpha ps:: save 0ps:: [1.0 0.0 0.0 1.0 12.33499 6.66562 ] concat ps:: pgfs ps::[end]color push gray 0gps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.30) cvn /DEST pdfmark end6fcolor push gray 0jSolutionk(a):  color popTfo!wardfacon!tradiction,ifBf-(BX) =BYz@1C0aBYz@2ffwithfBYz@1B;1Yz@2nonemptyfandopMeninBYn,thenr>Bf-zF@1 4(Bf-(BX)) EBXonfonehand,and{Bf-zF@1 4(Bf-(BX)) =BfzF@1 4(BYz@1)C0a BfzF@1(BYz@2)on$Othe$Pother. WIfBfR iscon!tinuous,thepreimages$PBf-zF@1 4(BYzCid)areopMen(andnonempt!y),soBX con!tains`aadisconnectedsubset.However,everysubsetofaaconnectedsetmustabMeconnected,softhiscon!tradictstheconnectednessofBX.) color pop ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]Gcolor push gray 0 color pop/color push gray 01L color popvgcolor push gray 0WExercises+-color push gray 0X9 color pop -ɍ냍color push gray 0ff color pop-ɍ냍color push gray 0f color popG color pop6@Oᎎ p+ᎎps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if endP ps:SDict begin H.S endps:SDict begin H.R endEps:SDict begin [/View [/XYZ H.V]/Dest (page.10) cvn /DEST pdfmark endɿ1덍color push gray 0iFC냍Tvgcolor push gray 0 =color push gray 0.6color push gray 0.6ٚ$_Ž color popcolor push gray 0LGcolor push gray 12 color popןٚ$ǎ_ color pop_ color pop3Exercises3]T color pop67냍Tcolor push gray 03]T color pop;+8 Tcolor push gray 03]T3]T color popG color pop{zps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save ps:: save ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 2ps:: [1.0 0.0 0.0 1.0 234.00299 32.44429 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 2ps:: [1.0 0.0 0.0 1.0 234.00299 32.44429 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: save &ps:: -16322.99998 -16322.99998 moveto %ps:: 16322.99998 -16322.99998 lineto $ps:: 16322.99998 16322.99998 lineto %ps:: -16322.99998 16322.99998 lineto ps:: closepath ps:: 0.0 3.83093 moveto ps:: 0.0 61.05763 lineto ps:: 0.99628 62.62318 2.26538 63.8923 3.83093 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466.73691 66.78307 468.00601 65.51395 468.00601 63.9484 curveto ps:: 468.00601 2.83466 lineto :ps:: 468.00601 1.2691 466.73691 0.0 465.17136 0.0 curveto ps:: 2.83466 0.0 lineto /ps:: 1.2691 0.0 0.0 1.2691 0.0 2.83466 curveto ps:: closepath *ps:: gsave pgffc pgffill grestore newpath ps:: restore ps:: save ps:: 3.9851 pgfw &color push rgb 0.15294 0.31374 0.43922ps:: 1.99255 66.78305 moveto ps:: 1.99255 0.0 lineto ps:: pgfstr color pop ps:: restore ps:: save !ps:: 1.0 .pgfsetfillopacityalpha ps:: save 0ps:: [1.0 0.0 0.0 1.0 11.33871 5.66934 ] concat ps:: pgfs ps::[end]color push gray 0Y3ps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.32) cvn /DEST pdfmark endps:SDict begin H.S endps:SDict begin 13.6 H.A endNps:SDict begin [/View [/XYZ H.V]/Dest (definition.2.0.5) cvn /DEST pdfmark end6fcolor push gray 0jDenition=u2.0.52(IdealfQuotien!t) color popFforft!woidealsBJz@1B;1Jz@2ffHEBR,theidepalquotientfisdenedbyBJz@1ʧ: BJz@2: p=n 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3.87196 moveto ;ps:: 5.58722 3.81293 5.68996 3.9043 5.68996 3.9043 curveto =ps:: 5.84854 4.05696 5.91335 4.33893 5.91335 4.33893 curveto =ps:: 5.62532 4.18321 5.46962 3.93054 5.52835 3.87196 curveto ps:: 0.28142 0.1518 moveto ps:: 0.72772 0.16045 lineto =ps:: 0.72772 0.16045 1.68256 1.19595 3.03905 1.83842 curveto =ps:: 4.39555 2.48108 5.88565 2.14183 6.09113 2.67725 curveto =ps:: 6.09113 2.67725 5.36833 2.60588 5.32368 2.92737 curveto ps:: 11.38336 5.77429 11.66884 5.97063 12.142 5.75655 curveto =ps:: 12.142 5.75655 12.68623 5.8727 13.14117 6.08665 curveto @ps:: 13.59659 6.30092 13.66507 6.37485 13.61276 6.53201 curveto @ps:: 13.61276 6.53201 12.61421 6.31517 11.21867 5.85126 curveto =ps:: 9.82362 5.38737 9.04927 5.11423 7.49675 4.43332 curveto =ps:: 5.94423 3.75275 3.62906 2.48442 2.49913 1.77747 curveto ;ps:: 1.36955 1.07065 0.28142 0.1518 0.28142 0.1518 curveto ps:: 2.50331 1.87411 moveto ps:: 9.17027 5.28336 10.31232 5.6402 11.65985 6.05081 curveto =ps:: 13.0077 6.46127 13.6054 6.55968 13.6054 6.55968 curveto >ps:: 13.6054 6.55968 13.59659 6.80965 13.09685 6.7562 curveto @ps:: 12.59709 6.70258 11.91876 6.26524 11.91876 6.26524 curveto >ps:: 12.16888 6.48814 12.16888 6.6757 12.16888 6.6757 curveto >ps:: 11.14221 6.40797 10.92809 6.14908 10.77672 5.926 curveto @ps:: 10.62451 5.70308 10.50883 5.73878 10.50883 5.73878 curveto >ps:: 10.39264 5.81892 10.49971 6.03305 10.7852 6.1313 curveto >ps:: 11.07098 6.22923 11.0532 6.57744 11.0532 6.57744 curveto ?ps:: 10.73174 6.26524 10.14285 6.01529 9.76808 5.88135 curveto =ps:: 9.39316 5.74727 9.34853 5.65828 9.34853 5.65828 curveto =ps:: 9.08975 5.11389 8.69691 5.09596 8.83966 5.36353 curveto =ps:: 8.98256 5.63155 9.29506 5.68532 9.29506 5.68532 curveto :ps:: 9.43782 5.8279 9.49126 6.1758 9.49126 6.1758 curveto ps:: 9.50919 5.89928 10.6871 6.38127 11.13356 6.76515 curveto ps:: 11.07082 6.36348 lineto @ps:: 11.07082 6.36348 11.63312 6.66705 12.32924 6.86342 curveto ps:: 12.222 6.55984 lineto ;ps:: 12.222 6.55984 13.31958 7.04181 13.659 6.6757 curveto =ps:: 13.99808 6.30988 12.84673 5.81 12.31097 5.67636 curveto ?ps:: 12.31097 5.67636 12.5254 5.56003 12.20409 5.23009 curveto >ps:: 11.88292 4.89993 11.01704 4.8017 11.01704 4.8017 curveto ?ps:: 11.01704 4.8017 11.24062 4.46262 10.59796 4.18576 curveto ;ps:: 9.9553 3.9091 9.34853 3.93597 9.34853 3.93597 curveto ;ps:: 9.34853 3.93597 9.4023 3.48056 8.7419 3.31989 curveto =ps:: 8.08115 3.15941 7.57262 3.23972 7.57262 3.23972 curveto =ps:: 7.57262 3.23972 7.59038 2.92737 7.16183 2.76686 curveto =ps:: 6.73346 2.60603 6.13562 2.65982 6.13562 2.65982 curveto =ps:: 6.13562 2.65982 6.09113 2.25815 5.05579 2.15094 curveto ps:: gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore ps:: newpath ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]ps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save ps:: save ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 3ps:: [1.0 0.0 0.0 1.0 234.00299 130.75162 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 3ps:: [1.0 0.0 0.0 1.0 234.00299 130.75162 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: save &ps:: -16322.99998 -16322.99998 moveto %ps:: 16322.99998 -16322.99998 lineto $ps:: 16322.99998 16322.99998 lineto %ps:: -16322.99998 16322.99998 lineto ps:: closepath ps:: 0.0 0.0 moveto ps:: 0.0 257.6723 lineto >ps:: 0.0 259.7881 1.71513 261.50325 3.83093 261.50325 curveto ps:: 464.17508 261.50325 lineto Gps:: 466.29086 261.50325 468.00601 259.7881 468.00601 257.6723 curveto ps:: 468.00601 0.0 lineto ps:: closepath ps:: clip ps:: newpath ps:: restore ps:: save .ps:: /pgffc{0.62746 0.76863 1 setrgbcolor}def !ps:: 1.0 .pgfsetfillopacityalpha ps:: 0.0 0.0 moveto ps:: 0.0 257.6723 lineto >ps:: 0.0 259.7881 1.71513 261.50325 3.83093 261.50325 curveto ps:: 464.17508 261.50325 lineto Gps:: 466.29086 261.50325 468.00601 259.7881 468.00601 257.6723 curveto ps:: 468.00601 0.0 lineto ps:: 467.00974 0.0 lineto ps:: 467.00974 257.6723 lineto Ips:: 467.00974 259.23787 465.74063 260.50697 464.17508 260.50697 curveto ps:: 3.83093 260.50697 lineto Bps:: 2.26538 260.50697 0.99628 259.23787 0.99628 257.6723 curveto ps:: 0.99628 0.0 lineto ps:: closepath ,ps:: gsave pgffc pgfeofill grestore newpath ps:: restore ps:: save ps:: /pgffc{0.95 setgray}def !ps:: 0.0 .pgfsetfillopacityalpha ps:: 0.99628 0.0 moveto ps:: 0.99628 257.6723 lineto Cps:: 0.99628 259.23785 2.26538 260.50696 3.83093 260.50696 curveto ps:: 464.17506 260.50696 lineto Hps:: 465.74062 260.50696 467.00972 259.23785 467.00972 257.6723 curveto ps:: 467.00972 0.0 lineto ps:: closepath *ps:: gsave pgffc pgffill grestore newpath ps:: restore ps:: save !ps:: 1.0 .pgfsetfillopacityalpha ps:: save 0ps:: [1.0 0.0 0.0 1.0 12.33499 7.57472 ] concat ps:: pgfs ps::[end]color push gray 0%ps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.36) cvn /DEST pdfmark endcolor push gray 0jSolution:  color popThatLBXdvE}BYisagainLananevdDariet!yfollowsfromwritingBX=ʓBVn(BI)B;bY9=BV(BJ ),XthenBXJEnBYy= BVn(BI+BJ )fwhereBIJ+BJHEޯBkX?[Bxz@1B;1E?B;1xzCnPB;yz@1B;E?B;yzCm].SofletWPBXJEnBYy= BUE[BVwithfBU';1VUpropMerandclosed,andletBCXB;CY #bMetheprojectionson!tothefactors.!ps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.37) cvn /DEST pdfmark endcolor push gray 0jClaim:U color popFforfeac!hBx E2BX,fBdzF@1 (Bx) E=BYUiscon!tainedinonlyoneofBUorBVn. Notefthatifthisistrue,w!ecanwriteBX= BGCU .E[nBGCV .wherev'dBGCU $鞞: t= n SBx E2BXC.^ ^ ^ ~BedF@1cCX (Bx)EBUiofare$the$pMoin!tsforwhic!htheen!tirebMerliesinBU1,DandsimilarlyBGCV {Harethoseforwhic!hthe bMer!liesinBVn.OIfw!ecanthenshowthatBGCUBB;1GCV xpare!closed,@ybyirreducibility!ofBXthiswillforcef(wlog)BGCV ao= E;andBX=BGCUB.ButthenW\3BedF@1cCX (BX) =BXJEnBYUand_BedF@1cCX(BGCUB)=BUF=,E)NBXJEnBYy=BU':whic!hfshowsthatBXJEnBYUisirreducible.) color pop ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]/color push gray 01L color popvgcolor push gray 0WExercises-color push gray 0X10 color pop -ɍ냍color push gray 0ff color pop-ɍ냍color push gray 0f color popG color pop6@Oᎎ +ᎎps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if endP ps:SDict 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curveto ps:: closepath ,ps:: gsave pgffc pgfeofill grestore newpath ps:: restore ps:: save ps:: /pgffc{0.95 setgray}def !ps:: 0.0 .pgfsetfillopacityalpha ps:: 0.99628 3.83093 moveto ps:: 0.99628 303.92232 lineto ps:: 467.00972 303.92232 lineto ps:: 467.00972 3.83093 lineto Cps:: 467.00972 2.26538 465.74062 0.99628 464.17506 0.99628 curveto ps:: 3.83093 0.99628 lineto =ps:: 2.26538 0.99628 0.99628 2.26538 0.99628 3.83093 curveto ps:: closepath *ps:: gsave pgffc pgffill grestore newpath ps:: restore ps:: save !ps:: 1.0 .pgfsetfillopacityalpha ps:: save 0ps:: [1.0 0.0 0.0 1.0 12.33499 6.66562 ] concat ps:: pgfs ps::[end]color push gray 0^4֍ps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save ps:: save ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 0ps:: [1.0 0.0 0.0 1.0 221.668 64.69992 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 0ps:: [1.0 0.0 0.0 1.0 221.668 64.69992 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: save &ps:: -16322.99998 -16322.99998 moveto %ps:: 16322.99998 -16322.99998 lineto $ps:: 16322.99998 16322.99998 lineto %ps:: -16322.99998 16322.99998 lineto ps:: closepath ps:: 0.0 3.83093 moveto ps:: 0.0 125.5689 lineto >ps:: 0.0 127.6847 1.71513 129.39984 3.83093 129.39984 curveto ps:: 439.50508 129.39984 lineto Gps:: 441.62088 129.39984 443.33603 127.6847 443.33603 125.5689 curveto ps:: 443.33603 3.83093 lineto ;ps:: 443.33603 1.71513 441.62088 0.0 439.50508 0.0 curveto ps:: 3.83093 0.0 lineto 1ps:: 1.71513 0.0 0.0 1.71513 0.0 3.83093 curveto ps:: closepath ps:: clip ps:: newpath ps:: restore ps:: save ps:: /pgffc{0.25 setgray}def "ps:: 0.25 .pgfsetfillopacityalpha ps:: 0.0 3.83093 moveto ps:: 0.0 125.5689 lineto >ps:: 0.0 127.6847 1.71513 129.39984 3.83093 129.39984 curveto ps:: 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439.50508 128.40356 lineto Hps:: 441.07065 128.40356 442.33975 127.13446 442.33975 125.5689 curveto ps:: 442.33975 3.83093 lineto Cps:: 442.33975 2.26538 441.07065 0.99628 439.50508 0.99628 curveto ps:: 3.83093 0.99628 lineto =ps:: 2.26538 0.99628 0.99628 2.26538 0.99628 3.83093 curveto ps:: closepath *ps:: gsave pgffc pgffill grestore newpath ps:: restore ps:: save !ps:: 1.0 .pgfsetfillopacityalpha ps:: save 0ps:: [1.0 0.0 0.0 1.0 12.33499 6.66562 ] concat ps:: pgfs ps::[end]color push gray 0fps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.38) cvn /DEST pdfmark end6feDcolor push gray 0Prpoof(Everybperiscontainedinoneirreduciblecomponent). color popFforfan!yxedBx,wecanwritepIBedF@1cCX (Bx) = BedF@1cCX(Bx)nE\BU>&GE[ngBedF@1cCX(Bx)nE\BV>)E B:SincepMoin!tsareclosedintheZariskitopologyandBCX ǃiscon!tinuous,veachBedF@1cCX (Bx)isclosed.andZth!usZBedF@1cCX (Bx)E\BUisclosed(andsimilarlyforBVn).ijNotingthatBedF@1cCX (Bx) E 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lineto Gps:: 441.07065 159.19867 442.33975 157.92957 442.33975 156.364 curveto ps:: 442.33975 3.83093 lineto Cps:: 442.33975 2.26538 441.07065 0.99628 439.50508 0.99628 curveto ps:: 3.83093 0.99628 lineto =ps:: 2.26538 0.99628 0.99628 2.26538 0.99628 3.83093 curveto ps:: closepath ,ps:: gsave pgffc pgfeofill grestore newpath ps:: restore ps:: save 4ps:: /pgffc{0.95294 0.88235 0.75294 setrgbcolor}def "ps:: 0.25 .pgfsetfillopacityalpha ps:: 0.99628 3.83093 moveto ps:: 0.99628 156.364 lineto Cps:: 0.99628 157.92957 2.26538 159.19867 3.83093 159.19867 curveto ps:: 439.50508 159.19867 lineto Gps:: 441.07065 159.19867 442.33975 157.92957 442.33975 156.364 curveto ps:: 442.33975 3.83093 lineto Cps:: 442.33975 2.26538 441.07065 0.99628 439.50508 0.99628 curveto ps:: 3.83093 0.99628 lineto =ps:: 2.26538 0.99628 0.99628 2.26538 0.99628 3.83093 curveto ps:: closepath *ps:: gsave pgffc pgffill grestore newpath ps:: restore ps:: save !ps:: 1.0 .pgfsetfillopacityalpha ps:: save 0ps:: [1.0 0.0 0.0 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5.84854 4.05696 5.91335 4.33893 5.91335 4.33893 curveto =ps:: 5.62532 4.18321 5.46962 3.93054 5.52835 3.87196 curveto ps:: 0.28142 0.1518 moveto ps:: 0.72772 0.16045 lineto =ps:: 0.72772 0.16045 1.68256 1.19595 3.03905 1.83842 curveto =ps:: 4.39555 2.48108 5.88565 2.14183 6.09113 2.67725 curveto =ps:: 6.09113 2.67725 5.36833 2.60588 5.32368 2.92737 curveto ps:: 11.38336 5.77429 11.66884 5.97063 12.142 5.75655 curveto =ps:: 12.142 5.75655 12.68623 5.8727 13.14117 6.08665 curveto @ps:: 13.59659 6.30092 13.66507 6.37485 13.61276 6.53201 curveto @ps:: 13.61276 6.53201 12.61421 6.31517 11.21867 5.85126 curveto =ps:: 9.82362 5.38737 9.04927 5.11423 7.49675 4.43332 curveto =ps:: 5.94423 3.75275 3.62906 2.48442 2.49913 1.77747 curveto ;ps:: 1.36955 1.07065 0.28142 0.1518 0.28142 0.1518 curveto ps:: 2.50331 1.87411 moveto ps:: 9.17027 5.28336 10.31232 5.6402 11.65985 6.05081 curveto =ps:: 13.0077 6.46127 13.6054 6.55968 13.6054 6.55968 curveto >ps:: 13.6054 6.55968 13.59659 6.80965 13.09685 6.7562 curveto @ps:: 12.59709 6.70258 11.91876 6.26524 11.91876 6.26524 curveto >ps:: 12.16888 6.48814 12.16888 6.6757 12.16888 6.6757 curveto >ps:: 11.14221 6.40797 10.92809 6.14908 10.77672 5.926 curveto @ps:: 10.62451 5.70308 10.50883 5.73878 10.50883 5.73878 curveto >ps:: 10.39264 5.81892 10.49971 6.03305 10.7852 6.1313 curveto >ps:: 11.07098 6.22923 11.0532 6.57744 11.0532 6.57744 curveto ?ps:: 10.73174 6.26524 10.14285 6.01529 9.76808 5.88135 curveto =ps:: 9.39316 5.74727 9.34853 5.65828 9.34853 5.65828 curveto =ps:: 9.08975 5.11389 8.69691 5.09596 8.83966 5.36353 curveto =ps:: 8.98256 5.63155 9.29506 5.68532 9.29506 5.68532 curveto :ps:: 9.43782 5.8279 9.49126 6.1758 9.49126 6.1758 curveto ps:: 9.50919 5.89928 10.6871 6.38127 11.13356 6.76515 curveto ps:: 11.07082 6.36348 lineto @ps:: 11.07082 6.36348 11.63312 6.66705 12.32924 6.86342 curveto ps:: 12.222 6.55984 lineto ;ps:: 12.222 6.55984 13.31958 7.04181 13.659 6.6757 curveto =ps:: 13.99808 6.30988 12.84673 5.81 12.31097 5.67636 curveto ?ps:: 12.31097 5.67636 12.5254 5.56003 12.20409 5.23009 curveto >ps:: 11.88292 4.89993 11.01704 4.8017 11.01704 4.8017 curveto ?ps:: 11.01704 4.8017 11.24062 4.46262 10.59796 4.18576 curveto ;ps:: 9.9553 3.9091 9.34853 3.93597 9.34853 3.93597 curveto ;ps:: 9.34853 3.93597 9.4023 3.48056 8.7419 3.31989 curveto =ps:: 8.08115 3.15941 7.57262 3.23972 7.57262 3.23972 curveto =ps:: 7.57262 3.23972 7.59038 2.92737 7.16183 2.76686 curveto =ps:: 6.73346 2.60603 6.13562 2.65982 6.13562 2.65982 curveto =ps:: 6.13562 2.65982 6.09113 2.25815 5.05579 2.15094 curveto ps:: gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore ps:: newpath ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end] hps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save ps:: save ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 3ps:: [1.0 0.0 0.0 1.0 234.00299 124.93623 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 3ps:: [1.0 0.0 0.0 1.0 234.00299 124.93623 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: save &ps:: -16322.99998 -16322.99998 moveto %ps:: 16322.99998 -16322.99998 lineto $ps:: 16322.99998 16322.99998 lineto %ps:: -16322.99998 16322.99998 lineto ps:: closepath ps:: 0.0 0.0 moveto ps:: 0.0 246.04153 lineto ?ps:: 0.0 248.15733 1.71513 249.87248 3.83093 249.87248 curveto ps:: 464.17508 249.87248 lineto Ips:: 466.29086 249.87248 468.00601 248.15733 468.00601 246.04153 curveto ps:: 468.00601 0.0 lineto ps:: closepath ps:: clip ps:: newpath ps:: restore ps:: save .ps:: /pgffc{0.62746 0.76863 1 setrgbcolor}def !ps:: 1.0 .pgfsetfillopacityalpha ps:: 0.0 0.0 moveto ps:: 0.0 246.04153 lineto ?ps:: 0.0 248.15733 1.71513 249.87248 3.83093 249.87248 curveto ps:: 464.17508 249.87248 lineto Ips:: 466.29086 249.87248 468.00601 248.15733 468.00601 246.04153 curveto ps:: 468.00601 0.0 lineto ps:: 467.00974 0.0 lineto ps:: 467.00974 246.04153 lineto Fps:: 467.00974 247.6071 465.74063 248.8762 464.17508 248.8762 curveto ps:: 3.83093 248.8762 lineto Aps:: 2.26538 248.8762 0.99628 247.6071 0.99628 246.04153 curveto ps:: 0.99628 0.0 lineto ps:: closepath ,ps:: gsave pgffc pgfeofill grestore newpath ps:: restore ps:: save ps:: /pgffc{0.95 setgray}def !ps:: 0.0 .pgfsetfillopacityalpha ps:: 0.99628 0.0 moveto ps:: 0.99628 246.04153 lineto @ps:: 0.99628 247.6071 2.26538 248.8762 3.83093 248.8762 curveto ps:: 464.17506 248.8762 lineto Gps:: 465.74062 248.8762 467.00972 247.6071 467.00972 246.04153 curveto ps:: 467.00972 0.0 lineto ps:: closepath *ps:: gsave pgffc pgffill grestore newpath ps:: restore ps:: save !ps:: 1.0 .pgfsetfillopacityalpha ps:: save 0ps:: [1.0 0.0 0.0 1.0 12.33499 6.66562 ] concat ps:: pgfs ps::[end]color push gray 0&ps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.41) cvn /DEST pdfmark endcolor push gray 0jSolution: q color popWfe'llfusethefollo!wingfactfromlinearalgebra:4 Fps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save ps:: save ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 0ps:: [1.0 0.0 0.0 1.0 221.668 24.37027 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 0ps:: [1.0 0.0 0.0 1.0 221.668 24.37027 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: save &ps:: -16322.99998 -16322.99998 moveto %ps:: 16322.99998 -16322.99998 lineto $ps:: 16322.99998 16322.99998 lineto %ps:: -16322.99998 16322.99998 lineto ps:: closepath ps:: 0.0 2.83466 moveto ps:: 0.0 45.9059 lineto ;ps:: 0.0 47.47145 1.2691 48.74055 2.83466 48.74055 curveto ps:: 440.50136 48.74055 lineto Eps:: 442.06693 48.74055 443.33603 47.47145 443.33603 45.9059 curveto ps:: 443.33603 2.83466 lineto :ps:: 443.33603 1.2691 442.06693 0.0 440.50136 0.0 curveto ps:: 2.83466 0.0 lineto /ps:: 1.2691 0.0 0.0 1.2691 0.0 2.83466 curveto ps:: closepath ps:: clip ps:: newpath ps:: restore ps:: save ps:: /pgffc{0.25 setgray}def "ps:: 0.15 .pgfsetfillopacityalpha ps:: 0.0 2.83466 moveto ps:: 0.0 45.9059 lineto ;ps:: 0.0 47.47145 1.2691 48.74055 2.83466 48.74055 curveto ps:: 440.50136 48.74055 lineto Eps:: 442.06693 48.74055 443.33603 47.47145 443.33603 45.9059 curveto ps:: 443.33603 2.83466 lineto :ps:: 443.33603 1.2691 442.06693 0.0 440.50136 0.0 curveto ps:: 2.83466 0.0 lineto /ps:: 1.2691 0.0 0.0 1.2691 0.0 2.83466 curveto ps:: closepath ps:: 0.0 2.83466 moveto ps:: 0.0 45.9059 lineto ;ps:: 0.0 47.47145 1.2691 48.74055 2.83466 48.74055 curveto ps:: 440.50136 48.74055 lineto Eps:: 442.06693 48.74055 443.33603 47.47145 443.33603 45.9059 curveto ps:: 443.33603 2.83466 lineto :ps:: 443.33603 1.2691 442.06693 0.0 440.50136 0.0 curveto ps:: 2.83466 0.0 lineto /ps:: 1.2691 0.0 0.0 1.2691 0.0 2.83466 curveto ps:: closepath ,ps:: gsave pgffc pgfeofill grestore newpath ps:: restore ps:: save 4ps:: /pgffc{0.15294 0.31374 0.43922 setrgbcolor}def "ps:: 0.15 .pgfsetfillopacityalpha ps:: 0.0 2.83466 moveto ps:: 0.0 45.9059 lineto ;ps:: 0.0 47.47145 1.2691 48.74055 2.83466 48.74055 curveto ps:: 440.50136 48.74055 lineto Eps:: 442.06693 48.74055 443.33603 47.47145 443.33603 45.9059 curveto ps:: 443.33603 2.83466 lineto :ps:: 443.33603 1.2691 442.06693 0.0 440.50136 0.0 curveto ps:: 2.83466 0.0 lineto /ps:: 1.2691 0.0 0.0 1.2691 0.0 2.83466 curveto ps:: closepath *ps:: gsave pgffc pgffill grestore newpath ps:: restore ps:: save ps:: 3.9851 pgfw &color push rgb 0.15294 0.31374 0.43922ps:: 1.99255 48.74054 moveto ps:: 1.99255 0.0 lineto ps:: pgfstr color pop ps:: restore ps:: save !ps:: 1.0 .pgfsetfillopacityalpha ps:: save 0ps:: [1.0 0.0 0.0 1.0 11.33871 7.79056 ] concat ps:: pgfs ps::[end]color push gray 0ܖfps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.42) cvn /DEST pdfmark endps:SDict begin H.S endps:SDict begin 13.6 H.A endNps:SDict begin [/View [/XYZ H.V]/Dest (definition.2.0.9) cvn /DEST pdfmark end6fcolor push gray 0jDenitionk(MatrixMinor) color popFforps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.44) cvn /DEST pdfmark end 6fcolor push gray 0jClaim:t% color popTheidealabMo!veisprime,andsothecoMordinateringBA(BX)isadomainandth!usBX isfirreducible.-sps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.45) cvn /DEST pdfmark endcolor push gray 0jClaim:U color popdimT(BX) =4. Heuristic:@therearethreedegreesoffreedominc!hoMosingtherstrowBx;1yd;z{I.ruTfoenforcetherank1condition,thesecondro!wmustbMeascalarm!ultipleoftherst,yieldingonedegreeoffreedomfforthescalar..&color push rgb 0.95294 0.93726 0.87843&color push rgb 0.95294 0.93726 0.87843 ) color popcolor push gray 0sYNote:I loGoked atthisforacoupleofhours,:butIdon'tknowhowtoproveeitherof stheseUUstatementswiththetoGolswehavesofar!) color pop) color pop) color pop ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]color push gray 0 color popps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.46) cvn /DEST pdfmark endps:SDict begin H.S endps:SDict begin 13.6 H.A endMps:SDict begin [/View [/XYZ H.V]/Dest (exercise.2.0.11) cvn /DEST pdfmark end34color push gray 0jExercise22.0.11k(Gathmann2.34):# color popLetfBX^bMeatopologicalspace,andsho!wps:SDict begin H.S endps:SDict begin 13.6 H.A endEps:SDict begin [/View [/XYZ H.V]/Dest (Item.12) cvn /DEST pdfmark endvhjcolor push gray 0{a. color pop`IffEf BUzCiНEgxjps:: 10.69238 5.82552 10.8247 6.08392 10.8247 6.08392 curveto >ps:: 10.5517 6.00443 10.4573 5.80759 10.57538 5.81638 curveto ps:: 8.95343 5.26703 moveto ;ps:: 9.10353 5.32288 9.2618 5.62242 9.2618 5.62242 curveto =ps:: 8.86557 5.46384 8.80365 5.21133 8.95343 5.26703 curveto ps:: 7.32346 4.70905 moveto ;ps:: 7.4614 4.7649 7.55548 5.07307 7.55548 5.07307 curveto ;ps:: 7.2764 4.92348 7.18536 4.6532 7.32346 4.70905 curveto ps:: 5.52835 3.87196 moveto ;ps:: 5.58722 3.81293 5.68996 3.9043 5.68996 3.9043 curveto =ps:: 5.84854 4.05696 5.91335 4.33893 5.91335 4.33893 curveto =ps:: 5.62532 4.18321 5.46962 3.93054 5.52835 3.87196 curveto ps:: 0.28142 0.1518 moveto ps:: 0.72772 0.16045 lineto =ps:: 0.72772 0.16045 1.68256 1.19595 3.03905 1.83842 curveto =ps:: 4.39555 2.48108 5.88565 2.14183 6.09113 2.67725 curveto =ps:: 6.09113 2.67725 5.36833 2.60588 5.32368 2.92737 curveto ps:: 11.38336 5.77429 11.66884 5.97063 12.142 5.75655 curveto =ps:: 12.142 5.75655 12.68623 5.8727 13.14117 6.08665 curveto @ps:: 13.59659 6.30092 13.66507 6.37485 13.61276 6.53201 curveto @ps:: 13.61276 6.53201 12.61421 6.31517 11.21867 5.85126 curveto =ps:: 9.82362 5.38737 9.04927 5.11423 7.49675 4.43332 curveto =ps:: 5.94423 3.75275 3.62906 2.48442 2.49913 1.77747 curveto ;ps:: 1.36955 1.07065 0.28142 0.1518 0.28142 0.1518 curveto ps:: 2.50331 1.87411 moveto ps:: 9.17027 5.28336 10.31232 5.6402 11.65985 6.05081 curveto =ps:: 13.0077 6.46127 13.6054 6.55968 13.6054 6.55968 curveto >ps:: 13.6054 6.55968 13.59659 6.80965 13.09685 6.7562 curveto @ps:: 12.59709 6.70258 11.91876 6.26524 11.91876 6.26524 curveto >ps:: 12.16888 6.48814 12.16888 6.6757 12.16888 6.6757 curveto >ps:: 11.14221 6.40797 10.92809 6.14908 10.77672 5.926 curveto @ps:: 10.62451 5.70308 10.50883 5.73878 10.50883 5.73878 curveto >ps:: 10.39264 5.81892 10.49971 6.03305 10.7852 6.1313 curveto >ps:: 11.07098 6.22923 11.0532 6.57744 11.0532 6.57744 curveto ?ps:: 10.73174 6.26524 10.14285 6.01529 9.76808 5.88135 curveto =ps:: 9.39316 5.74727 9.34853 5.65828 9.34853 5.65828 curveto =ps:: 9.08975 5.11389 8.69691 5.09596 8.83966 5.36353 curveto =ps:: 8.98256 5.63155 9.29506 5.68532 9.29506 5.68532 curveto :ps:: 9.43782 5.8279 9.49126 6.1758 9.49126 6.1758 curveto ps:: 9.50919 5.89928 10.6871 6.38127 11.13356 6.76515 curveto ps:: 11.07082 6.36348 lineto @ps:: 11.07082 6.36348 11.63312 6.66705 12.32924 6.86342 curveto ps:: 12.222 6.55984 lineto ;ps:: 12.222 6.55984 13.31958 7.04181 13.659 6.6757 curveto =ps:: 13.99808 6.30988 12.84673 5.81 12.31097 5.67636 curveto ?ps:: 12.31097 5.67636 12.5254 5.56003 12.20409 5.23009 curveto >ps:: 11.88292 4.89993 11.01704 4.8017 11.01704 4.8017 curveto ?ps:: 11.01704 4.8017 11.24062 4.46262 10.59796 4.18576 curveto ;ps:: 9.9553 3.9091 9.34853 3.93597 9.34853 3.93597 curveto ;ps:: 9.34853 3.93597 9.4023 3.48056 8.7419 3.31989 curveto =ps:: 8.08115 3.15941 7.57262 3.23972 7.57262 3.23972 curveto =ps:: 7.57262 3.23972 7.59038 2.92737 7.16183 2.76686 curveto =ps:: 6.73346 2.60603 6.13562 2.65982 6.13562 2.65982 curveto =ps:: 6.13562 2.65982 6.09113 2.25815 5.05579 2.15094 curveto ps:: gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore ps:: newpath ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]hzps: ps::[begin] ps:: pgfo ps:: save ps:: 0 setgray ps:: 0.3985 pgfw ps:: save ps:: restore ps:: save ps:: restore ps:: save ps:: save ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 3ps:: [1.0 0.0 0.0 1.0 234.00299 172.29562 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: save ps:: 0.0 pgfw ps:: save ps:: 0.0 pgfw ps:: restore ps:: save 3ps:: [1.0 0.0 0.0 1.0 234.00299 172.29562 ] concat ps:: pgfs ps::[end]ps: 0 setgray ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: save &ps:: -16322.99998 -16322.99998 moveto %ps:: 16322.99998 -16322.99998 lineto $ps:: 16322.99998 16322.99998 lineto %ps:: -16322.99998 16322.99998 lineto ps:: closepath ps:: 0.0 0.0 moveto ps:: 0.0 340.76031 lineto ?ps:: 0.0 342.87611 1.71513 344.59126 3.83093 344.59126 curveto ps:: 464.17508 344.59126 lineto Ips:: 466.29086 344.59126 468.00601 342.87611 468.00601 340.76031 curveto ps:: 468.00601 0.0 lineto ps:: closepath ps:: clip ps:: newpath ps:: restore ps:: save .ps:: /pgffc{0.62746 0.76863 1 setrgbcolor}def !ps:: 1.0 .pgfsetfillopacityalpha ps:: 0.0 0.0 moveto ps:: 0.0 340.76031 lineto ?ps:: 0.0 342.87611 1.71513 344.59126 3.83093 344.59126 curveto ps:: 464.17508 344.59126 lineto Ips:: 466.29086 344.59126 468.00601 342.87611 468.00601 340.76031 curveto ps:: 468.00601 0.0 lineto ps:: 467.00974 0.0 lineto ps:: 467.00974 340.76031 lineto Ips:: 467.00974 342.32588 465.74063 343.59499 464.17508 343.59499 curveto ps:: 3.83093 343.59499 lineto Cps:: 2.26538 343.59499 0.99628 342.32588 0.99628 340.76031 curveto ps:: 0.99628 0.0 lineto ps:: closepath ,ps:: gsave pgffc pgfeofill grestore newpath ps:: restore ps:: save ps:: /pgffc{0.95 setgray}def !ps:: 0.0 .pgfsetfillopacityalpha ps:: 0.99628 0.0 moveto ps:: 0.99628 340.76031 lineto Cps:: 0.99628 342.32588 2.26538 343.59499 3.83093 343.59499 curveto ps:: 464.17506 343.59499 lineto Ips:: 465.74062 343.59499 467.00972 342.32588 467.00972 340.76031 curveto ps:: 467.00972 0.0 lineto ps:: closepath *ps:: gsave pgffc pgffill grestore newpath ps:: restore ps:: save !ps:: 1.0 .pgfsetfillopacityalpha ps:: save 0ps:: [1.0 0.0 0.0 1.0 12.33499 6.66562 ] concat ps:: pgfs ps::[end]color push gray 0ps:SDict begin H.S endps:SDict begin 13.6 H.A endTps:SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.47) cvn /DEST pdfmark endcolor push gray 0jSolution: color pop&color push rgb 0.95294 0.93726 0.87843&color push rgb 0.95294 0.93726 0.87843C4) color popcolor push gray 04sYStrictlyUUfornotationalconvenience,UUwe'lltreat,"Ŏ lmsy10fUVUn0erlmmi7ni}gYisifitwereacountableopGencover.) color pop) color 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`K(BAz@1,i.e.thattheinclusionsarestillpropMer,fw!ecanjustnotethat>›Bx E2BAzCi@+10EnnBAzCi z(E(,)NBxE2(LBIzCi@+1E\nBUz@02)9tEnn(BIzCiE\nBUz@0'Jʝ).=(LBIz@1.EnnBIz@2$Ŀ)+uE\nBUz@0ʧ=E;B:wButfthisexhibitsalengthBnc!haininBUz@0,sodimUBUz@0ʧE Bn.Tfakingsuprema,wehave\Bn Edim2BUz@0ʧEsup C0NCiF2CJZ0dim)BUzCio=Bs:) color pop ps::[begin] ps:: pgfr ps:: restore ps:: restore ps:: restore ps:: newpath ps:: restore ps:: pgfc ps::[end]/color push gray 01L color popvgcolor push gray 0WExercises-color push gray 0X12 color pop -ɍ냍color push gray 0ff color pop-ɍ냍color push gray 0f color popG color pop6@Oᎎ Ďᎎps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if endP ps:SDict begin H.S endps:SDict begin H.R endEps:SDict begin [/View [/XYZ H.V]/Dest (page.13) cvn /DEST pdfmark endɿ1덍color push gray 0iFC냍Tvgcolor 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