Algebra general exam. August 17 2021, 9am -1pm

Your UVa ID Number:

Directions.

• 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.

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

Problem 1

(12 pts) Let $n \in \mathbb{N}$ and let $A=\left(a_{i j}\right) \in \operatorname{Mat}_{n}(\mathbb{Q})$ be the matrix given by $a_{i j}=i$ for $1 \leq i, j \leq n$, that is,

$$A=\left(\begin{array}{cccc} 1 & 1 & \ldots & 1 \\ 2 & 2 & \ldots & 2 \\ \ldots & \ldots & \ldots & \ldots \\ n & n & \ldots & n \end{array}\right)$$

Compute
(a) The characteristic polynomial of $A$
(b) The minimal polynomial of $A$
(c) The Jordan canonical form of $A$
(d) The rational canonical form of $A$

Problem 2

Given a group $G$, denote by $\operatorname{Aut}(G)$ its group of automorphisms.
(a) ( 7 pts) Let $G=\mathbb{Z}_{p^{a}} \oplus \mathbb{Z}_{p^{b}}$ where $a \geq b>0$. Show that $\operatorname{Aut}(G)$ is non-abelian by explicitly constructing two noncommuting automorphisms. Hint: It may be helpful to start with the case $a=b=1$.
(b) ( 8 pts) Let $G$ be a finite abelian group. Show that $\operatorname{Aut}(G)$ is abelian $\Longleftrightarrow G$ is cyclic. Hint: Use (a). Remember that you are allowed to do this even if you did not solve (a).

Problem 3

(a) ( 8 pts) Let $p$ be a prime and $G$ a group or order $p^{3}$. Prove that any two elements $x, y$ of $G$ which are conjugate in $G$ commute (that is, $x y=y x)$. Hint: Prove that any element $g$ of $G$ is contained in some abelian normal subgroup of $G$ (which depends on $g$ ).
(b) ( 7 pts ) Give, with arguments, an example of a group $G$ of order 16 and two elements $x, y$ of $G$ which are conjugate in $G$ but do not commute.

Problem 4

( 14 pts ) Let $R$ be a (commutative) UFD (with 1 ), let $f \in R$, and assume that $f$ is non-unit and nonzero. Let $R_{f}=R\left[\frac{1}{f}\right]$ (you can think of $R$ as the ring of fractions of $R$ with the set of denominators $D=\left\{1, f, f^{2}, \ldots\right\}$ or as the subring of the field of fractions $\operatorname{Frac}(R)$ generated by $R$ and $\frac{1}{f})$. Prove that

$$R_{f}^{\times} \cong R^{\times} \times \mathbb{Z}^{m} \text { for some } m \in \mathbb{N}$$

(Here $S^{\times}$is the multiplicative group of $S$ ). Give a detailed argument.

Problem 5

Let $R$ be a commutative ring with 1 and let $M$ and $N$ be finitely generated $R$-modules.
(a) ( 5 pts) Prove that $M \otimes_{R} N$ is finitely generated.
(b) ( 9 pts) Assume in addition that $M$ is Noetherian. Prove that $M \otimes_{R} N$ is Noetherian. Note: You can use standard properties of Noetherian modules (unless they are equivalent or almost equivalent to the statement of (b)), but state clearly what you are using.

Problem 6

Consider $f(x)=x^{4}+a x^{2}-1 \in \mathbb{Q}[x]$ where $a \in \mathbb{Z} \backslash\{0\}$. Let $K$ be the splitting field of $f(x)$ over $\mathbb{Q}$.
(a) ( 5 pts) Show that $f(x)$ is irreducible
(b) ( 4 pts) Show that $i \in K$
(c) ( 7 pts) Determine the isomorphism class of the Galois group $\operatorname{Gal}(K / \mathbb{Q})$ (your answer should be of the form $\operatorname{Gal}(K / \mathbb{Q}) \cong G$ where $G$ is a familiar group).

Problem 7

(a) ( 7 pts) Let $n \in \mathbb{N}$, let $F$ be an algebraically closed field either of characteristic 0 or of characteristic $p>0$ where $p$ does not divide $n$. Prove that $F$ contains a primitive $n^{\text {th }}$ root of unity.
(b) ( 7 pts) According to (a) $\overline{\mathbb{F}}_{3}$ (the algebraic closure of a field with 3 elements) contains a primitive $13^{\text {th }}$ root of unity, call it $\omega$. Compute $\left[\mathbb{F}_{3}(\omega): \mathbb{F}_{3}\right]$ (with proof).