Algebra general exam

January 12, 2017

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

Consider the polynomial $f(X)=X^{4}-2 X^{2}-6$. Prove this polynomial is irreducible. Describe the splitting field of this polynomial (including its degree over $\mathbb{Q}$ ), and the Galois group of this splitting field (hint: pay attention to which roots are real and which are complex). (15pt)

Problem 2

Consider a field $K$, and two finite extensions $L, M$ of $K$. Consider the $K$-algebra $L \otimes_{K} M$ (with the usual multiplication $(a \otimes b)(c \otimes d)=a c \otimes b d$ ). Prove that $L \otimes_{K} M$ is a field if and only if any time an extension $E / K$ contains subfields $L^{\prime}$ and $M^{\prime}$ isomorphic to $L$ and $M$, the composite $L M$ has degree $[L M: K]=[L: K][M: K]$. (15pt)

Problem 3

Let $R$ be a commutative ring, and $M, N$ be $R$-modules. Show that for any submodules $M^{\prime} \subset M$ and $N^{\prime} \subset N$, the induced map $M \otimes_{R} N \rightarrow\left(M / M^{\prime}\right) \otimes_{R}\left(N / N^{\prime}\right)$ has kernel given by $M^{\prime} \otimes N+M \otimes N^{\prime}$ (hint: use the universal property of tensor products). (10pt)

Problem 4

We call a group $G$ polycyclic if it contains a series of subgroups $\{e\}=G_{0} \subset G_{1} \subset G_{2} \subset \cdots \subset G_{n}=G$ such that $G_{i} / G_{i-1}$ is a (possibly infinite) cyclic group.
(a) Show that a finite group is polycyclic if and only if it is solvable. (5pt)
(b) Show that $\mathbb{Q}$ is an example of an abelian group which is not polycyclic. (5pt)

Problem 5

Given a finite group $G$ and two subgroups $H, K$, the double cosets of $H$ and $K$ are the sets of the form $H g K$ for some $g \in G$.
(a) Show that any two double cosets must be equal or disjoint. (5pt)
(b) Show that the size of any double coset must divide the product of the orders $\# H \cdot \# K$. (5pt)
(c) Find an example of a double coset whose size does not divide the order $\# G$. (5pt)

Problem 6

(a) Find the smallest integer $n$ such that $S_{n}$ has a subgroup of order 10 , but $S_{k}$ for $k<n$ does not. (5pt)
(b) Find the smallest integer $m$ such that $S_{m}$ has an element of order 10, but $S_{k}$ for $k<m$ does not. (5pt)

Problem 7

Consider the matrix

$$A=\left[\begin{array}{ccc} 12 & 4 & -16 \\ 4 & 3 & -7 \\ 8 & 3 & -11 j \end{array}\right]$$

(a) Find the characteristic and minimal polynomials of this polynomial and its Jordan normal form. (8pt)
(b) Consider map $\mathbb{Z}^{3} \rightarrow \mathbb{Z}^{3}$ induced by $A$. Describe the kernel and cokernel of this map as a sum of copies of $\mathbb{Z}$ and $\mathbb{Z} / n \mathbb{Z}$. (7pt)

Problem 8

Let $A$ be the ring of $n \times n$ matrices over a field $F$.
(a) Show the right ideals of $A$ are precisely the subsets of the form

$$\{X \in A \mid \operatorname{image}(X) \subset V\}$$

where $V$ ranges over all linear subspaces of $F^{n}$. (5 pts)
(b) Show the left ideals of $A$ are precisely the subsets of the form

$$\{X \in A \mid \operatorname{kernel}(X) \supset W\}$$

where $W$ ranges over all linear subspaces of $F^{n}$. (5 pts)
(c) Show that $A$ is a simple ring: its only 2 -sided ideals are $A$ itself, and $\{0\}$. (5 pts)