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date: 2023-02-11 21:50
title: Videos Kedlaya NT
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  - Videos Kedlaya NT
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created: 2023-02-11T21:50
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# Videos Kedlaya NT

Kedlay NT1 https://www.youtube.com/watch?v=nOhzhK-eT4I&list=PLoWHl5YajIf5i41NHooPWSANY2-8urmpB

Kedlaya NT2 https://www.youtube.com/watch?v=KVQ8h0Vg43Y&list=PLoWHl5YajIf7HgNP934VAwp1kMCNb0_Ki

# NT1

## Lectures 1 and 2

- [ ] What are $\bar \ZZ$ and $\bar \QQ$?
	- [ ] $\exists p\in \ZZ[x]$  (resp $\QQ[x]$) such that $p(\alpha) = 0$.
- [ ] Give an example of a $\alpha\not\in \bar{\ZZ}$.
	- [ ] $\alpha \da i/2$, since its minimal polynomial is $x^2 + {1\over 4}\not\in\ZZ[x]$.
- [ ] What is the cardinality of $\bar \ZZ$?
	- [ ] Countable
- [ ] Why is $\ZZ[i]$ Euclidean?
	- [ ] For $a,b\in \ZZ[i]$, take $a/b\in \CC$ and round to the nearest lattice point.
	- [ ] Distance is ${r\over  b} \da d \leq {1\over \sqrt 2}< 1$, so yields ${a\over b} = q + {r\over b}\implies a=qb+r$ where $\abs{r} < \abs{b}$.
- [ ] Describe the Euclidean algorithm.
	- [ ] $r_0 = a$
	- [ ] $r_1 = b$
	- [ ] $r_2$ is the remainder of $r_0/r_1$
	- [ ] $r_3$ is the remainder of $r_1/r_2$
	- [ ] Then $(a,b) =(r_0, r_1) = (r_1, r_2) = \cdots  = (r_{k-1}, 0) = r_{k-1}$ since $r_0 - r_2 = \ell r_1$.
- [ ] What are the units in $\ZZ[i]$?
	- [ ] $\ts{\pm 1,\pm i} = \ts{x\in \ZZ[i] \st N(x) = 1}$.
- [ ] What are the primes in $\ZZ[i]$?
	- [ ] $1+i$
	- [ ] $a+bi$ where $N(a+bi) = p \equiv 1\mod 4$.
	- [ ] $p$ where $p \equiv 3 \mod 4$.
	- [ ] Follows because $N(\alpha) = p$ prrime forces $\alpha$ to be prime.
- [ ] Which primes are sums of two squares?
	- [ ] $p\equiv 1 \mod 4$ and $p=2$.
- [ ] Give a congruence for $(p-1)!$.
	- [ ] $(p-1)! \equiv -1 \mod p \iff p$ is prime.
- [ ] Discuss properties of $\ZZ[i]$.
	- [ ] PID
	- [ ] Euclidean domain

## Lecture 3

- [ ] What is $\zeta_3$ in rectangular coordinates?
	- [ ] $\zeta_3 = {1\over 2}(-1 + \sqrt{3} i)$.
- [ ] What is $\ff\ZZ[\zeta_3]$?
	- [ ] $\QQ(\zeta_3) = \QQ(\sqrt{-3})$.
- [ ] What are the units in $\ZZ[\zeta_3]$?
	- [ ] $\ts{\pm 1, \pm \zeta_3, \pm\zeta_3^2}$.
- [ ] What are the primes in $\ZZ[\zeta_3]$?
	- [ ] $1-\zeta_3$ with norm 3
	- [ ] $N(a+b\zeta_3) = p \equiv 1\mod 3$
	- [ ] $p\equiv 2 \mod 3$.
- [ ] For $K = \QQ(\sqrt{-D})$, what is $\ZZ_K$?
	- [ ] $\ZZ_K = \ZZ\adjoin{\sqrt{-D}}$ if $D\not\equiv 1 \mod 4$.
	- [ ] $\ZZ_K = \ZZ\adjoin{1 + \sqrt{D} \over 2}$ if $D\equiv 1 \mod 4$.
- [ ] Which imaginary quadratic fields are PIDs? Which are EDs?
	- [ ] Only 9 cases: $-1,-2,-3,-7,-11,-19,-43,-67,-163$.
	- [ ] EDs: the first five.
	- [ ] Note: onlly finitely many units.
- [ ] Which real quadratic fields are PIDS? EDs?
	- [ ] EDs: Finitely many.
	- [ ] PIDs: unknown, conjecturally infinitely many.
	- [ ] Infinitely many units: $a^2-b^2D=1$ is the Pell equation and has infinitely many solutions.
- [ ] Give an example of failure of unique factorization.
	- [ ] $6 = (2)(3) = (1+\sqrt{-5})(1-\sqrt{-5})\in \ZZ\adjoin{\sqrt{-5}}$, all factors are irreducible.

## Lecture 4