ALGEBRA GENERAL EXAM JANUARY 10, 2004

Problem 1

( 10 point) Let $G=\left\{g_{1}, \cdots, g_{n}\right\}$ a finite abelian group. Show that the product $P:=g_{1} \cdots g_{n}$ is of order 1 or 2 .

Problem 2

(10 point) Let $H$ be the subgroup of $G:=\mathbb{Z} \oplus \mathbb{Z}=\{(m, n) \mid m, n \in \mathbb{Z}\}$ generated by $(1,2)$ and $(3,4)$. Identify the quotient group $G / H$.

Problem 3

( 15 point)

(a) If $A \in M_{n}(\mathbb{R})$ is idempotent, i.e. $A^{2}=A$, then it is diagonalizable.
(b) Two idempotent matrices $A, B \in M_{n}(\mathbb{R})$ are similar if and only if they have the same rank.

Problem 4

( 10 point) Show that a principal ideal in $\mathbb{Z}[x]$ can never be maximal.

Problem 5

( 15 point) Let $\xi$ be a primitive 9 -th root of unity. Let $K=\mathbb{Q}(\xi)$ and $F=\mathbb{Q}\left(\xi+\xi^{-1}\right)$.
(a) Show that $[K: F]=2$.
(b) Show that the extension $F / Q$ is normal.

Problem 6

Let $V$ and $W$ be finite dimensional vector spaces over a field $k$, and let $f: V \rightarrow V$ and $g: W \rightarrow W$ be linear operators. One can define a linear operator $f \otimes g: V \otimes_{k} W \rightarrow V \otimes_{k} W$. Show that $\operatorname{Tr}(f \otimes g)=\operatorname{Tr} f \cdot \operatorname{Tr} g$.

Problem 7

( 15 point) Let $S$ be a finite set acted upon by a finite group $G$. Denote by $\mathbb{C}(S)$ the complex vector space of complex-valued functions on $S$.
(a) Show that the map $G \times \mathbb{C}(S) \rightarrow \mathbb{C}(S),(x, f) \mapsto x . f$, defines a $G$-action on $\mathbb{C}(S)$. Here $x . f \in \mathbb{C}(S)$ is defined by $x . f(s)=f\left(x^{-1} s\right)$ for all $s \in S$.
(b) Show that the dimension of the subspace of $G$-fixed points in $\mathbb{C}(S)$ is equal to the number of $G$-orbits in $S$.

Problem 8

( 15 point) TRUE-FALSE. You do not need to justify your answers.
(a) Every matrix in $S L_{n}(\mathbb{R})$ has a positive real eigenvalue if $n$ is odd.
(b) The Galois group of $x^{3}+1$ is isomorphic to $S_{3}$.
(c) A group of order 99 must have a unique normal subgroup of order 11.
(d) There exists a unique finite field of order 6 up to isomorphism.
(e) $\mathbb{Z}[x]$ is a Euclidean domain.