Algebra General Exam - August 2022

Your UVa ID Number:

• Please show all your work and justify any statements that you make.

• State clearly and fully any theorem you use.

• Vague statements and hand-waving arguments will not be appreciated.

• You may assume the statement in an earlier part proven in order to do a later part.

DO EACH PROBLEM ON A SEPARATE SHEET OF PAPER, AND STAPLE THEM TOGETHER IN THE CORRECT ORDER BEFORE TURNING THE EXAM IN.

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"On my honor, I pledge that I have neither given nor received help on this assignment."

Problem 1

Let $R$ denote the commutative ring $\mathbb{Z}[\sqrt{-5}]=\{a+b \sqrt{-5} \mid a, b \in \mathbb{Z}\}$.
(a) ( 6 points) Construct two nonassociated factorizations of 6 into products of irreducible elements (you need to verify that all elements involved in these factorizations are indeed irreducible and the two factorizations are indeed nonassociated).
(b) ( 5 points) Using the factorizations from (a), construct an ideal $\mathfrak{a} \subset R$ which is not principal.
(c) ( 6 points) Show that the square $\mathfrak{a}^{2}=\mathfrak{a a}$ of the ideal from part (b) is principal.

Problem 2

For a complex $n \times n$-matrix $A=\left(a_{i j}\right)$ we let $\bar{A}$ denote the matrix $\left(\bar{a}_{i j}\right)$ where the bar denotes the complex conjugation. Also, we let $\chi_{A}$ denote the characteristic polynomial of $A$.
(a) (4 points) If $A$ and $\bar{A}$ are conjugate, prove that the characteristic polynomial $\chi_{A}$ has real coefficients.
(b) ( 7 points) Conversely, if $A$ is diagonalizable and $\chi_{A}$ has real coefficients then $A$ and $\bar{A}$ are conjugate.
(c) ( 5 points) Give an example of a nondiagonalizable complex matrix $A$ such that $\chi_{A}$ has real coefficients but $A$ and $\bar{A}$ are not conjugate.

Problem 3

(a) (7 points) Let $G$ be a group, and $H \subset G$ be a normal subgroup. Assume that the center $Z(H)$ is $\{e\}$ and that every automorphism of $H$ is inner. Show that $G$ is the direct product $H \times C_{G}(H)$ where $C_{G}(H)$ is the centralizer of $H$ in $G$. (Hint. Consider the action of $G$ on $H$ by inner automorphisms.)
(b) ( 8 points) Show that there is no group $G$ such that the commutator subgroup $[G, G]$ is isomorphic to the symmetric group $S_{3}$. (You can assume without proof that $S_{3}$ satisfies the assumptions on $H$ made in part (a).)

Problem 4

Let $R$ be a commutative ring with identity. We say that a finitely generated $R$ module $P$ is projective if $P$ is a direct summand of a free $R$-module. That is, there is some $d \geq 1$ and another $R$-module $Q$ such that $P \oplus Q \simeq R^{d}$ as $R$-modules.
(a) ( 6 points) Suppose $P$ is a finitely generated projective $R$-module. Show that for every epimorphism of $R$-modules $\varphi: M \rightarrow N$, an arbitrary $R$-module homomorphism $\alpha: P \rightarrow N$ lifts to a homomorphism $\beta: P \rightarrow M$ such that $\varphi \circ \beta=\alpha$.
(b) ( 5 points) Show that if $P_{1}$ and $P_{2}$ are finitely generated projective modules then their tensor product $P_{1} \otimes_{R} P_{2}$ is also a finitely generated projective $R$-module.

Problem 5

(a) ( 7 points) Let $V$ and $W$ be vector spaces over a field $K$, and let $v_{1}, \ldots, v_{n} \in V$ be linearly independent vectors. Show that if $w_{1}, \ldots, w_{n} \in W$ are such that $v_{1} \otimes w_{1}+\cdots+v_{n} \otimes w_{n}=0$ in $V \otimes_{K} W$ then $w_{1}=\cdots=w_{n}=0$.
(b) ( 7 points) Again, let $V$ and $W$ be vector spaces over a field $K$, and let $x \in V \otimes_{K} W$. If $x=v_{1} \otimes w_{1}+\cdots+v_{n} \otimes w_{n}$ is a shortest presentation of $x$ as a sum of simple tensors (i.e., $x$ cannot be written as $v_{1}^{\prime} \otimes w_{1}^{\prime}+\cdots+v_{m}^{\prime} \otimes w_{m}^{\prime}$ with $m<n$ ) then the vectors $v_{1}, \ldots, v_{n}$ (and likewise $w_{1}, \ldots, w_{n}$ ) are linearly independent.
Note: Make sure you clearly state which properties of the tensor product you are using in both parts of this problem.

Problem 6

( 8 points) Let $\alpha$ be a complex number satisfying $\alpha^{6}+3=0$. Show that $\mathbb{Q}(\alpha) / \mathbb{Q}$ is a Galois extension, and determine its Galois group. (Hint. The same is not true if 3 is replaced by 2 .)

Problem 7

Assume that $p$ is a prime number and $A \in G L_{5}\left(\mathbb{F}_{p}\right)$ is a matrix that satisfies $A^{3}=1$.
(a) ( 4 points) If $p \equiv 1 \bmod 3$, show that $A$ is diagonalizable.
(b) ( 7 points) Let $p=11$. Classify all conjugacy classes of such matrices A.

Problem 8

(8 points) Let $K$ be a subfield of $\mathbb{R}$, let $f(x) \in K[x]$ be an irreducible polynomial, and let $L$ be the splitting field of $f$ over $K$ (i.e., the field obtained by adjoining to $K$ all complex roots of $f$ ). Assume that the Galois group $\operatorname{Gal}(L / K)$ is abelian. Show that if one root of $f$ is real then all roots are real.