Algebra general exam

Your name:

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

• State any theorem you use clearly and fully.

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

• 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

1. ( 10 points) Classify, up to isomorphism, all finite groups of order $2 p$, where $p$ is a prime number.

Problem 2

2. ( 12 points, 6 points each) Consider the ring
   $R=\mathbb{Z}[\sqrt{-11}]=\{m+n \sqrt{-11} \mid m, n \in \mathbb{Z}\}$.
   (a) Is $R$ a UFD? Give arguments for your answer.
   (b) Exhibit an ideal $I$ in $R$ which is not principal. Show that your $I$ is not principal.

Problem 3

3. ( 15 points, 5 points each) Decide in each of the following three cases whether the given polynomial is irreducible. Include arguments.
   (a) $x^{2}-2 i$ in $\mathbb{Z}[i][x]$;
   (b) $x^{3}-49 x^{2}+(3+\sqrt{2}) x+7$ in $\mathbb{Z}[\sqrt{2}][x]$;
   (c) $x^{2}+x y+y^{2}$ in $\mathbb{C}[x, y]$.

Problem 4

4. ( 12 points) Let $A$ be a finite abelian group of order $n, p$ a prime divisor of $n$ and $n=p^{k} m$ with $k, m \in \mathbb{N}$ such that $(p, m)=1$. Denote by $A_{p}$ the Sylow $p$-subgroup of A .
   (a) ( 8 points) Show that the abelian groups $A_{p}$ and $\mathbb{Z} / p^{k} \mathbb{Z} \otimes_{\mathbb{Z}} A$ are isomorphic.
   (b) ( 4 points) Describe $\mathbb{Z} / p \mathbb{Z} \otimes_{\mathbb{Z}} A$ as an abelian group without using tensor products but (certain) invariants of $A$.

Problem 5

5. (16 points) We set $M:=M_{3}(\mathbb{Q})$ and denote by 0 the zero matrix and by $I$ the identity matrix of $M$.
   (a) (2 points) Prove or disprove: If $A \in M$ satisfies $A^{6}=0$, then also $A^{3}=0$.
   (b) ( 4 points) Classify, up to similarity, all matrices in $M$ satisfying $A^{6}=0$. Exhibit one representative for each such similarity class.
   (c) (2 points) Prove or disprove: If $A \in M$ satisfies $A^{6}=I$, then also $A^{3}=I$.
   (d) (8 points) Classify, up to similarity, all matrices in $M$ satisfying $A^{6}=I$. Exhibit one representative for each such similarity class.

Problem 6

6. (10 points) Let $n \geq 2$ be a natural number, $F$ a field and $A=\left(a_{i j}\right) \in M_{n}(F)$ the matrix with entries $a_{i j}=j \cdot 1_{F} \in F$ for all $1 \leq i, j \leq n$.
   Determine the characteristic polynomial, the minimal polynomial and the JCF of $A$.
   Hint: The result may depend on the characteristic of $F$.

Problem 7

7. (15 points)
   (a) ( 8 points) Construct, using cyclotomic fields, a Galois extension $K$ of $\mathbb{Q}$ of degree 3. Include arguments.
   (b) (7 points) Find, explicitly, a polynomial $f(x) \in \mathbb{Q}[x]$ such that your field $K$ in (a) is the splitting field of $f(x)$ over $\mathbb{Q}$.

Problem 8

8. (10 points) Let $M|K| F$ be a tower of finite field extensions such that $M \mid F$ and $K \mid F$ are both Galois. Assume that the Galois group $G(M \mid K)$ is cyclic. Prove that $L \mid F$ is Galois for every intermediate field $L$ with $M|L| K$.