Algebra General Exam - January 2023

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• Please Prove 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

(a) ( 8 points) Let $G$ be a finite group, and $H \varsubsetneqq G$ be a proper subgroup. Prove that

$$G \neq \bigcup_{g \in G} g H g^{-1}$$

(Hint. Estimate the number of elements in the right-hand side.)
(b) ( 8 points) Let $G$ be a transitive subgroup of the symmetric group $S_{n}$. Prove that there exists $g \in G$ that has no fixed points on $I_{n}=\{1,2, \ldots, n\}$ (i.e. there is no $i \in I_{n}$ such that $g(i)=i$ ).

Problem 2

Let $R=\mathbb{Z}[\sqrt[3]{2}]=\left\{a_{0}+a_{1} \sqrt[3]{2}+a_{2}(\sqrt[3]{2})^{2} \mid a_{0}, a_{1}, a_{2} \in \mathbb{Z}\right\}$.
(a) ( 5 points) Prove that $R$ is a subring of $\mathbb{R}$.
(b) (6 points) Prove that the evaluation homomorphism $\mathbb{Z}[x] \rightarrow R, p(x) \mapsto p(\sqrt[3]{2})$, yields a ring isomorphism

$$R \simeq \mathbb{Z}[x] /\left(x^{3}-2\right)$$

(c) ( 6 points) Using the isomorphism from part (b) and the 3rd Isomorphism Theorem, can you determine if the ideals $5 R$ and $7 R$ are prime or maximal?

Problem 3

(8 points) Let $F$ be a field (possibly finite). Prove that the polynomial ring $F[x]$ has infinitely many prime ideals.
Note: You cannot assume without proof that $F[x]$ has infinitely many irreducible polynomials.

Problem 4

(10 points) Let $R$ be a commutative ring with 1 . Let $f: \mathbb{Z}^{n} \rightarrow \mathbb{Z}^{m}$ be a $\mathbb{Z}$-module homomorphism, and let $K=\operatorname{ker} f$. Prove that there exists a submodule $M \subset \mathbb{Z}^{n}$ such that
$\mathbb{Z}^{n}=K \oplus M$. (Hint. Argue that $P:=\operatorname{im} f \subset \mathbb{Z}^{m}$ is projective.) Reminder. An $R$-module $P$ is called projective if, for any surjective module homomorphism $\varphi: M \rightarrow N$, every module homomorphism $\psi: P \rightarrow N$ admits a lifting $\tilde{\psi}: P \rightarrow M$ such that $\varphi \circ \tilde{\psi}=\psi$.

Problem 5

Let $L / K$ be a finite Galois extension of degree $d$ with Galois group $G$. In this problem, we think of the elements of $G$ as linear transformations $\sigma: L \rightarrow L$ of the $d$-dimensional vector space $L$ over $K$.
(a) ( 5 points) Let $\sigma \in G$ be an element of order $n$. Prove that the minimal polynomial of $\sigma: L \rightarrow L$ is $\mu(x)=x^{n}-1$. (Hint. Use the standard theorem about linear independence of characters.)
(b) ( 10 points) Assume that $p$ is a prime $\neq$ char $K$, and $K$ contains a primitive $p$-th root of unity. Let $L / K$ be a Galois extension of degree $p$, and let $\sigma \in \operatorname{Gal}(L / K)$ be a nontrivial automorphism. Prove that $\sigma$ is diagonalizable. Identify its eigenvalues, the corresponding eigenvectors and derive from this analysis that $L=K(\sqrt[p]{a})$ for some $a \in K^{\times}$. (Note. This gives an easier proof of Kummer theory in this special case).

Problem 6

( 10 points) Let $K / \mathbb{Q}$ be a Galois extension with Galois group $S_{n}$ (symmetric group) for some $n \geq 5$. Assume that $K$ contains a primitive root of unity $\zeta_{d}$. Prove that $d \leq 6$.

Problem 7

(a) (6 points) Let $L / K$ be a field extension and $f(x) \in K[x]$. Prove that there is an isomorphism of $L$-algebras

$$L \bigotimes_{K} \frac{K[x]}{f(x) K[x]} \simeq \frac{L[x]}{f(x) L[x]}$$

Note. The $L$-algebra structure on the ring $L \otimes_{K} \frac{K[x]}{f(x) K[x]}$ is given on simple tensors by $c \cdot(a \otimes \overline{g(x)}):=(c a) \otimes \overline{g(x)}$, where $c a$ is the ring multiplication in $L$.
(b) ( 7 points) Suppose that $L / K$ is a finite Galois extension of degree $n$. Prove that there is an isomorphism of $L$-algebras

$$L \otimes_{K} L \simeq \underbrace{L \times \cdots \times L}_{n} .$$

Problem 8

Let $V$ be the $\mathbb{C}$-vector space of all polynomials $p(x, y)$ in two variables of total degree $\leq 2$. Let $T: V \rightarrow V$ be the linear transformation $T(f)=\frac{\partial f}{\partial x}$.
(a) ( 2 points) Prove that $T$ is nilpotent.
(b) ( 9 points) Construct a Jordan basis for $T$ and compute its Jordan canonical form.