ALGEBRA GENERAL EXAM

August 20, 2018

Your UVa ID Number:

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

• State any theorem you use clearly and fully.

• Vague statements and unclear arguments will not be accepted as progress towards a solution.

• You may assume the statement of an earlier question proven in order to solve a later one.

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

Problem 1

Let $G$ be a group, and let $\gamma_{i}(G)$ denote the $i^{\text {th }}$ term of the lower central series of $G$. That is, $\gamma_{1}(G)=G$ and inductively $\gamma_{i+1}(G)= \left[G, \gamma_{i}(G)\right]$. Let $\operatorname{IAut}(G)<\operatorname{Aut}(G)$ be the group of automorphisms of $G$ which induce the trivial automorphism of $G /[G, G]$. Prove the following statements.
(a) (4 points) For each $i \geq 1$, we have $\gamma_{i}(G)<G$ is characteristic and $\gamma_{i}(G) / \gamma_{i+1}(G)$ is abelian.
(b) ( 6 points) If $\phi \in \operatorname{IAut}(G)$ and $i \geq 1$, then $\phi$ induces the trivial automorphism of $\gamma_{i}(G) / \gamma_{i+1}(G)$. You may use without proof the fact that for any group $G$, there is an inclusion

$$\left[[G, G], \gamma_{i}(G)\right] \subset \gamma_{i+2}(G)$$

Problem 2

( 10 points) Let $G$ be a non-cyclic group of order 57 . Determine the number of elements of all orders of $G$.

Problem 3

Let $R$ be a commutative ring such that all prime ideals are finitely generated. Prove that $R$ is Noetherian by completing the following steps.
(a) ( 5 points) Let $X$ be the set of non-finitely generated ideals of $R$. Prove that if $R$ is not Noetherian then $X$ has a maximal element, say $I$.
(b) (1 point) Prove that there exist elements $x, y \in R$ such that $x, y \notin I$ but such that $x y \in I$.
(c) (3 points) Prove that $I_{x}=I+R x$ and $J_{x}=\{r \in R \mid r x \in I\}$ are finitely generated ideals. Here, $x$ is the same element as in the previous part.
(d) ( 6 points) Conclude that $I$ is finitely generated and derive a contradiction.

Problem 4

Let $R$ be an integral domain and let $M$ be a nontrivial torsion $R$ module.
(a) ( 5 points) If $M$ is finitely generated then the annihilator of $M$ in $R$ is nontrivial. Recall that the annihilator of $M$ is the ideal $\{r \mid r m=0$ for all $m \in M\}$.
(b) ( 5 points) Find an integral domain $R$ and a torsion module $M$ over $R$ whose annihilator is the zero ideal.

Problem 5

Let $F$ be a field and let $n$ be a natural number.
(a) ( 10 points) Let $F=\mathbb{R}$. Classify the matrices $A \in M_{n}(\mathbb{R})$ satisfying $A^{3}=A$, up to similarity. That is to say, exhibit a matrix in each similarity class satisfying $A^{3}=A$.
(b) ( 5 points) For an appropriate $F$ and $n$, find a matrix $A \in M_{n}(F)$ which is not diagonalizable and which satisfies $A^{3}=A$.

Problem 6

(13 points) Let $p(x)$ be a polynomial defined over $\mathbb{R}$, and suppose that $p$ takes on rational values at rational numbers. Prove that the coefficients of $p$ are rational (Hint: use the Vandermonde determinant). Does the statement remain true if the rationals are replaced by the integers?

Problem 7

( 14 points) What is the Galois group of $x^{5}-1$ over a field with 7 elements?

Problem 8

(13 points) Describe the splitting field of $x^{4}+x^{2}+1$ over $\mathbb{Q}$. That is, describe elements of the algebraic closure of $\mathbb{Q}$ which need to be adjoined in order to obtain the splitting field. What is the degree of the splitting field over $\mathbb{Q}$ ?