ALGEBRA GENERAL EXAM AUGUST 16, 2003, UVA

Notations. Throughout the exam, we denote by $\mathbb{Z}_{n}$ the cyclic group of $n$ elements, $\mathbb{Q}$ the rational field, $\mathbb{R}$ the real field.

Problem 1

( 10 points) Let $G=S L_{2}(\mathbb{R})$ be the group of real $2 \times 2$ matrices with determinant one, and let $\mathbb{H}=\{z \in \mathbb{C} \mid \operatorname{Im} z>0\}$ be the upper half of the complex plane. It is known that the map $G \times \mathbb{H} \rightarrow \mathbb{H},(g, z) \mapsto g(z)$, defines an action of $G$ on $\mathbb{H}$, where

$$g(z)=\frac{a z+b}{c z+d}, \quad \text { for } g=\left[\begin{array}{cc} a & b \\ c & d \end{array}\right] \in G .$$

(a) Identify the stabilizer $G(i)$ of the point $z=i$ (the imaginary root) in $\mathbb{H}$ under this action.
(b) Show that the action of $G$ on $\mathbb{H}$ is transitive.

Problem 2

( 10 points) Let $D$ be a PID with $F$ as its field of fractions. Show that every element $x \in F$ can be written as a sum of primary fractions (i.e. with denominators powers of primes):

$$x=\sum_{i=1}^{n} \frac{a_{i}}{p_{i}^{e_{i}}}$$

for some $a_{1}, \ldots, a_{n} \in D$, and distinct primes $p_{1}, \ldots, p_{n} \in D$.

Problem 3

( 15 points) Let $P_{2 n-1}$ be the vector space of polynomials in one variable $x$ with real coefficients of degree $\leq 2 n-1$. Let $T$ denote the linear operator on $P_{2 n-1}$ defined by $p(x) \mapsto p(x)+p^{\prime \prime}(x)$ for every $p(x) \in P_{2 n-1}$, where $p^{\prime \prime}$ denotes the second derivative.
(a) Write the matrix $A_{T}$ of the operator $T$ with respect to the basis $\left\{1, x, x^{2}, \ldots, x^{2 n-1}\right\}$ of $P_{2 n-1}$.
(b) Find the Jordan canonical form of $A_{T}$ AND a Jordan basis for $T$.

Problem 4

( 15 points)
(a) Determine the following tensor products and explain your answers.
(i) $\mathbb{R} \bigotimes_{\mathbb{Z}} \mathbb{Z}_{2003}$
(ii) $\mathbb{Q}[x] /\left(x^{2}+1\right) \bigotimes_{\mathbb{Q}[x]} \mathbb{Q}[x] /\left(x^{2}+2\right)$.
(b) Let $V$ and $W$ be finite-dimensional vector spaces over a field $F$, and $\left\{v_{1}, \ldots, v_{n}\right\}$ be a basis of $V$. Prove that if

$$v_{1} \otimes w_{1}+\cdots+v_{n} \otimes w_{n}=0$$

in $V \bigotimes_{F} W$ for $w_{1}, \ldots, w_{n} \in W$, then $w_{1}=\cdots=w_{n}=0$.

Problem 5

( 10 points) Choose your favorite one, denoted by $G$, between the Klein four group (i.e. $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ ) and the diheral group $D_{4}$ of order 8. Provide an example of an irreducible degree 4 polynomial whose Galois group over $\mathbb{Q}$ is isomorphic to $G$. Show your work.

Problem 6

( 15 points) Let $B$ be a symmetric bilinear form on a finite-dimensional vector space $V$ over a field $F$. For a subspace $W \subset V$, we define the annihilator subspace of $W$ in $V: W^{\perp}=\{x \in V \mid B(x, w)=0$ for every $w \in W\}$. Assume further that $B$ is nondegenerate, that is $V^{\perp}=\{0\}$. Show that:
(a) $\operatorname{dim} W^{\perp}=\operatorname{dim} V-\operatorname{dim} W$.
(b) $\left(W^{\perp}\right)^{\perp}=W$.

Problem 7

(10 points for (a) and (b)) (Hint: proof by contradiction could be useful)

Let $G$ be a finite group of order $n$. Suppose that for every $d$ dividing $n$, the equation $x^{d}=1$ has at most $d$ solutions in $G$. Show that:
(a) For each prime $p$, the Sylow $p$-subgroup of $G$ is unique, and thus is normal.
(b) The Sylow $p$-subgroup of $G$ is cyclic.
(c) (bonus problem) Use (a) and (b) to show that $G$ is cyclic.

Problem 8

( 15 points) Determine whether each of the following statements is TRUE or FALSE. You do NOT need to show your work.
(a) Let $G$ be a finite group. Assume that $H$ is a normal subgroup of $G$ and $K$ is a normal subgroup of $H$. Then $K$ is normal in $G$.
(b) If $R$ is PID, so is $R[x]$.
(c) The integer orthogonal group $O_{n}(\mathbb{Z})$, which consists of $n \times n$ orthogonal matrices whose entries are all integers, is a finite group.
(d) Let $p$ be a prime. Then the number of monic irreducible polynomials of degree 2 over a finite field $\mathbb{F}_{p}$ is $p(p-1) / 2$.
(e) If an arbitrary integer polynomial $f(x)$ has no roots in $\mathbb{Z}$, then it has no roots in $\mathbb{Q}$.