Algebra II - August 2022

Instructions

The exam will last 2 hours. You will be graded on 4 out of the 5 questions. If you submit solutions to more than 4, you must state which question you choose for me to ignore on the first page of your solutions.

Problem 1.

Let $F=\mathbb{Q}(\sqrt[4]{3},i)$.

1. Identify isomorphism class of the Galois group $G$ of $F/\mathbb{Q}$ in terms of the classification of small groups.
2. Find all intermediate fields $F/K/\mathbb{Q}$ with $[K:\mathbb{Q}]=2$.

Problem 2.

Let $F$ be a field of characteristic $p>0$, such that $f(X)=X^p-X+a\in F[X]$ does not have a root in $F$. Show that $f(X)$ is irreducible, and that the Galois group of $f$ is cyclic of order $p$.

Hint: if $\alpha$ is a root of $f$ in an extension of $F$, show that $\alpha +1$ is also a root of $f$.

Problem 3.

1. Show that if $M$ and $N$ are projective $R$-modules, then $M{\otimes }_{R}N$ is projective.
2. Show that if $M$ and $N$ are injective $\mathbb{Z}$-modules then $M{\otimes }_{\mathbb{Z}}N$ is injective. Hint: This does not hold if $\mathbb{Z}$ were replaced by a general ring $R$.

Problem 4.

Show that ${\mathrm{Tor}}_i^{\mathbb{Z}/p^3\mathbb{Z}}(\mathbb{Z}/p\mathbb{Z},\mathbb{Z}/p^2\mathbb{Z})\cong \mathbb{Z}/p\mathbb{Z}$ for all $i\ge 0$.

Problem 5.

There is a group of order 24 called $S{L}_2({\mathbb{F}}_3)$. The table below contains the sizes of the conjugacy classes of $S{L}_2({\mathbb{F}}_3)$, and two of its irreducible complex characters. Use these characters to compute the entire complex character table of $S{L}_2({\mathbb{F}}_3)$.

Here $\xi$ is a primitive cube root of unity.