# Exam Review **Sections for the exam:** - 2.1-2.4 ($\mathbf R, \mathbf Q, \mathbf C$) and quadratic/cubic formulas - 3.1-3.3 (Euclidean algorithm, roots, $\mathbf Z[x]$) - 5.1 (Vector Spaces, toward field extensions) **Some unsolicited advice**: - First and foremost, review homeworks and problem sets. These are the most likely topics to show up on an exam. You'll want to know the statements of any results proved in exercises, along with the details of how you proved or computed things. - For every definition or theorem, have an example and a counterexample in mind! E.g. when defining a ring, know by heart a good example of a set that *is* a ring and a set that is *not* a ring. - Whenever possible, ask yourself "What's the picture?" I.e., is there some sketch/doodle/cartoon you can associate to a definition, theorem, or example that helps intuitively capture what's going on? Or if nothing else, you can use such pictures as mnemonic devices to help trigger recall later. - Know the definitions by heart! You'll want to have them memorized. - When reviewing proofs, try to remember a "sketch" of the idea first, i.e. a high-level understanding of the various steps or tricks used to move through the proof. A good example is exercise 3.3.8 on proving "Wilson's theorem" using field theory -- a sketch might look like - Trick: consider this very special polynomial $f(x) = x^p - x$. - Show it has $p$ distinct roots in $\mathbf{Z}/p\mathbf Z$ using the fundamental theorem of algebra and Fermat's Little Theorem - Factor out $f$ and remove the $(x-0)$ term to get $g(x) = {f(x) \over (x-0)} = x^{p-1} - 1 = (x-1)(x-2)\cdots$. - Plug in $g(p)$ and see what happens. - Some of these computations can be done by online calculators like WolframAlpha, Symbolab, Sagemath, etc. Do use these to check your work -- but also make sure you have tricks and techniques for doing it if you were stranded on a desert island with no electronics! (I.e. in an exam setting). **Notation**: - $R$ is usually a ring, and $A, B ,S$ etc are subsets of $R$. - $\mathbf{Z, Q, R, C}$ denote the fields of integers, rationals, reals, and complex numbers respectively. - The notation $:=$ means "this is a definition". - $\mathbf{Z}/m\mathbf{Z}$ is the ring of integers modulo $m$, and its elements are often denoted $\left\{\bar 0, \bar 1,\cdots, \overline{m-1}\right\}$. - $K, L$ are abstract fields (which could for example be $\mathbf{Q}$) and $\bar K, \bar L$ denote algebraic closures. - $K[\alpha]$ is the field extension of $K$ obtained by adjoining an element $\alpha\in \bar K$ to the base field $K$. - $R[x]$ is the ring of polynomials with coefficients in $R$, so finite degree polynomials of the form $a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0$ where $a_0, \cdots, a_n \in R$. - $i$ always means the purely imaginary complex number $e^{i\pi\over 2}$, which is a solution to the equation $x^2=-1$. - $\zeta_n := e^{2 i \pi \over n}$ is a *primitive $n$th root of unity*, one of the $n$ complex solutions to the equation $x^n=1$. - $\mathrm{M}_n(R), \mathrm{Mat}_n(R)$ denotes $n\times n$ matrices over a ring $R$. Note that such matrices may be singular. - $\mathrm{GL}_n(R)$ denotes the set of $n\times n$ invertible matrices with entries from $R$, i.e. matrices $M$ with $\det M \neq 0$. ## Rings and fields - Define what it means for a set $R$ to be a **ring**. - Define what it means for a subset $S \subseteq R$ to be a **subring**. - Let $A$ and $B$ be subrings of a ring $R$. Is $A \cap B$ a subring? Explain. - *Hint: true. How do you prove it?* - Let $A$ and $B$ be subrings of a ring $R$. Is $A \cup B$ a subring? Explain. - *Hint: false. Consider $a+b$ where $a\in A, b\in B$.* - Let $A$ and $B$ be subrings of a ring $R$. Is $A+B := \{ a+b\mid a\in A, b\in B\}$ a subring? Explain. - *Hint: false, consider $(a_1 + b_1)(a_2 + b_2)$*. - Prove that $M_2(\mathbf Z)$, the set of 2 by 2 matrices with integer coefficients, is a ring. - Prove that $GL_2(\mathbf{Z}) := \left\{ M\in M_2(\mathbf Z) \mid \det M \neq 0 \right\}$ is not a ring. - Define an **integral domain**. - Define a **zero divisor**. - Prove that $\mathbf{Z}/5\mathbf{Z}$ is an integral domain. - Prove that $\mathbf{Z}/6\mathbf{Z}$ is not an integral domain. - Prove that if $R$ is an integral domain, then its polynomial ring $R[x]$ is also an integral domain. (See p. 84) - Prove that $\mathbf{Z}/n\mathbf{Z}$ is an integral domain $\iff n$ is prime. - Let $R$ be an integral domain and assume that $a b=a c$, where $a, b, c \in R$ and $a \neq 0$. Prove that $b=c$. - Define a **field**. - Define a **unit** in a ring. - Prove that if $R$ is a field then $R$ is an integral domain. - Prove that $\mathbf{Z}/5\mathbf{Z}$ is a field by showing the multiplicative inverse of every element exists in the ring. - Prove that $\mathbf{Z}/6\mathbf{Z}$ is not a field by finding an element without a multiplicative inverse. - Let $R = \mathbf{Z}/m\mathbf{Z}$ for $m$ arbitrary. When is an element $\bar a\in R$ a unit? - Define an **ordered field** (see p. 47). - Prove that $\mathbf{Q}$ is an ordered field (see p.48) - Prove that $\sqrt 2$ is irrational. - Define the **least upper bound** or **supremum** of an ordered set (see p. 52). - (Slightly hard) Sketch a proof that $\sqrt 2 \in \mathbf{R}$ using the least upper bound property of the real numbers. - Prove that $\mathbf{Q}[\sqrt 2] := \left\{ a_0 \cdot 1 + a_1 \cdot \sqrt{2} \mid a_0, a_1\in \mathbf{Q} \right\}$ is a field (see p.54). *Hint: it suffices to find the multiplicative inverse of $a_0 + a_1\sqrt{2}$ and show it is still an element of $\mathbf{Q}[\sqrt 2]$*. - Prove that $\mathbf{Q}[2^{1\over 3}]$ is a field. *Hint: find a basis and write it as a set as above.* - Prove that $\mathbf{Q}[\sqrt d]$ is a field for any $d\in\mathbf{Z}$. - Write down all of the elements in $$R=\left\{\left(\begin{array}{ll}a & b \\ 0 & a\end{array}\right) \mid a, b \in \mathbf{Z}/2\mathbf Z\right\} \subset M_2\left(\mathbf{Z}/2\mathbf{Z}\right)$$. Is $R$ commutative? What are the additive and multiplicative inverses of all elements? Is $R$ an integral domain? Is $R$ a field? ## Complex numbers and field extensions - Let $R$ be a ring and $f\in R[x]$ a polynomial. Define what it means for $f$ to be **irreducible** in $R[x]$. - State the **rational root test**. - State **Eisenstein's criterion**. - Let $f\in \mathbf{Z}[x]$ be a polynomial with integer coefficients. How does the irreducibility of $f\in \mathbf{Z}[x]$ relate to the reducibility of $\bar f\in {\mathbf{Z} \over p\mathbf{Z}}[x]$? How can you use this to prove or disprove irreducibility of a polynomial? *Hint: there is a statement like "if $f$ is reducible over $\mathbf{Z}$, then $\bar f$ is ??? when reduced mod $p$".* - Let $K$ be a field and $a \in K$ by any element. Show that $f(x)\in K[x]$ is irreducible iff $g(x) := f(x+a)\in K[x]$ is irreducible. - Use this to conclude that $f(x)\in \mathbf{Q}[x]$ is irreducible iff $f(x+1) \in \mathbf{Q}[x]$ is irreducible. - Checking irreducibility: - Prove that $f(x) = x^2-2$ is irreducible in $\mathbf{Q}[x]$. - Prove that if $f\in R[x]$ has a root $\alpha\in R$, then $f$ is reducible in $R[x]$. - Prove that $x^4-5x^2+6$ has no rational roots $\alpha\in \mathbf{Q}$, but is still reducible in $\mathbf{Q}[x]$. - Show that $p(x)=x^3+x+1$ is irreducible in ${\mathbf Z \over 2\mathbf Z}[x]$. - Let $\mathbf{F}=\mathbf{Z}/2\mathbf{Z}$ and factor the polynomial $x^8+x \in \mathbf{F}[x]$ into irreducible factors. - Prove that $h(x)=x^4+2 x^2+5 x+1$ is irreducible over $\mathbf{Q}$ by reducing mod $p$ for some prime $p$. - Prove that the polynomial $f(x)=1+x+x^3+x^4$ is reducible over *any* field $\mathbf{F}$. - Let $R$ be a ring and $f, g\in R[x]$ be two polynomials. Define what the **greatest common divisor (GCD)** of $f$ and $g$ is, usually written $d(x) = \gcd(f(x), g(x))$. - Suppose that $f(x)=x^{99}-5 x^{49}+1 \in \mathbf{Q}[x]$. Compute the remainder when $f(x)$ is divided by - $g_1(x) = x$; - $g_2(x) = x-1$; - $g_3(x) = x+1$. - State **Euler's formula for complex numbers**. *Hint: this is the one that converts $e^{i\theta}$ to cosines and sines.* - Convert $5+5i$ to polar coordinates - $7e^{i\pi\over 3}$ to rectangular coordinates. - Let $z = re^{i\theta}$ and $w = \rho e^{i\phi}$, and prove that $z w=r \rho(\cos (\theta+\phi)+i \sin (\theta+\phi))$. - Prove de Moivre's theorem: $(\cos(x) +i \sin(x))^n=\cos(n x)+ i \sin (n x)$ - Practice factoring: - Factor $f(x) = x^n-1$ into irreducible polynomials. - Factor $f(x) = x^n-a$. *Hint: you'll need a term like $a^{1\over n}$. Try some concrete examples like $a=2, 3,4,\cdots$.* - Factor $f(x) = x^n+1$ - Factor $f(x) = x^n + a$. - Factoring over the complex numbers: - Plot the complex roots of $x^3-1$ in $\mathbf{C}$. - Plot the complex roots of $x^3+1$ in $\mathbf{C}$. - Plot the complex roots of $x^4-3$ in $\mathbf{C}$. - Plot the complex roots of $x^4+3$ in $\mathbf{C}$ - Perform the division algorithm to show the following (See p.85): $$ {x^3+2 \over 2x^2 + x + 1}= ({1\over 2}x - {1\over 4}) + {-{1\over 4}x + {9\over 4}\over 2x^2 + x + 1} \implies x^3 + 2 = ({1\over 2}x - {1\over 4})(2x^2+x+1) + (-{1\over 4}x + {9\over 4}) $$ - Define the gcd of two polynomials. How do you compute it? - Let $d(x) = \gcd(x^3-8, x^2-x-2)$ and show $d(x) = x-2$. Find polynomials $a(x), b(x) \in \mathbf{Q}[x]$ such that $$ d(x) = a(x)f(x) + b(x) g(x) $$ and by a computation, show that $$ d(x) = {1\over 3}f(x) + (-{1 \over 3 })(x+1)g(x) $$ - Find the greatest common divisor of the following polynomials over $\mathbf{Q}$. - $x^2+x-2$ and $x^5-x^4-10 x^3+10 x^2+9 x-9$ - $x^2+1$ and $x^6+x^3+x+1$. ## Splitting Fields and Extensions - State the **fundamental theorem of algebra** over an arbitrary algebraically closed field $K$. - Prove **Wilson's theorem**: if $p$ is prime, then $(p-1)! \equiv -1 \mod p$. Hint: use the very special polynomial $f(x) := x^p - x \in {\mathbf Z\over p\mathbf Z}[x]$. - Let $K$ be a field and define what it means for a set $V$ to be a **vector space** over $K$. - Define what it means for a subset $S\subseteq V$ of a vector space to be **linearly independent**. Define what it means for $S$ to be a **spanning set**. - Define what it means for a subset $S \subseteq V$ of a vector space to be a **basis** of $V$. - Define the **dimension** of a vector space $V$. - Prove that $\mathbf{C}$ is a 2-dimensional vector space over the field $\mathbf{R}$. - Prove that $\mathbf{R}^n$ is an $n$-dimensional vector space over $\mathbf R$. - Let $V = \left\{ a_0 + a_1 x + a_2 x^2 \mid a_i\in \mathbf R \right\} \subseteq \mathbf R[x]$ be the subset of all polynomials with real coefficients of degree at most 2. Prove that $V$ is a vector space over $\mathbf R$. What is its dimension? - Let $S\subseteq \mathbf{R}^3$ be the plane defined by $S := \left\{ (x,y,z)\in \mathbf{R}^3 \mid x+y+z = 1\right\}$. Prove that $S$ is not a vector subspace of $\mathbf{R}^3$. - Prove that $\mathrm{GL}_2(\mathbf R)$ is not a vector space. Hint: - Let $f\in \mathbf{Q}[x]$ and define what the **splitting field** of $f$ is. - Find the splitting field of $f(x) = x^3-1$. - Find the splitting field of $f(x) = (x^2-2)(x^2+1)$. - Find the splitting field of $f(x) = x^2+x+1$. - Let $L$ be a field extension of $K$, and define the **degree** of $L$ over $K$, denoted $[L : K]$. - What are the degrees of $\mathbf{Q}[\sqrt d]$ over $\mathbf Q$ for $d=2,3,4,5,6,7$? - What is the degree of $\mathbf C$ over $\mathbf R$? - Let $\alpha \in \mathbf{C}$ and define what the **minimal polynomial** of $\alpha$ is. - How does the degree of the minimal polynomial of $\alpha$ relate to the dimension of $\mathbf{Q}[\alpha]$ as a vector space over $\mathbf{Q}$? - Find the minimal polynomial of $\sqrt{d}$ for any $d\in \mathbf{Z}$ (positive or negative). - Find the minimal polynomials of $\zeta_3, \zeta_4, \zeta_5$. *Hint: these are not $x^3-1, x^4-1, x^5-1$ respectively! There are even smaller polynomials they satisfy.* - Show $\mathbf{Q}[\sqrt{3} + i] = \mathbf{Q}[\sqrt 3, i]$ by explicitly showing each field is contained in the other. *Hint: it's enough to show the generators of one are contained in the other and vice versa.* - For any $a, b\in \mathbf{Z}$ with $a,b>0$, prove that $\mathbf{Q}[\sqrt a, \sqrt b] = \mathbf{Q}[\sqrt a + \sqrt b]$. *Hint: you may assume $a\neq b$; prove subset containment both ways.* - Find a *power basis* for $K = \mathbf{Q}[\sqrt 3, \sqrt 7]$, which is defined to be an element $\alpha\in K$ such that the powers $\left\{\alpha^0, \alpha^1,\cdots, \alpha^n \right\}$ form a basis for $K$ as a vector space over $\mathbf{Q}$. *Hint: $K = \mathbf{Q}[\alpha]$ for $\alpha = \sqrt 3 + \sqrt 7$ by a previous problem. What is the minimal polynomial of $\alpha$? What is the dimension of $K$?* - Find a basis for - $\mathbf Q[i\sqrt7]$ over $\mathbf Q$. - $\mathbf Q[\sqrt{-3}]$ over $\mathbf Q$. - $\mathbf Q [i\sqrt d]$ over $\mathbf Q$, where $d\in \mathbf{Z}$ is a *squarefree* positive integer (i.e. if $d = \prod_{i=1}^m p_i^{e_i}$ is the prime factorization of $d$, then every $e_i = 1$, so $d$ is not divisible by any perfect square). - $\mathbf Q[\sqrt d + i]$ over $\mathbf Q \sqrt d$.