Algebra Qual 2014-09

Let $G$ be a finite abelian group of odd order. Let $\phi:G\to G$ be the function defined by $\phi(g)=g^2$ for all $g\in G$. Prove that $\phi$ is an automorphism.

Let $G$ be a group and suppose $H\leq G$. The {\bfseries normalizer} of $H$ in $G$ is defined to be $N(H)=\{g\in G\,|\, gH=Hg\}$ and the {\bfseries centralizer} of $H$ in $G$ is defined to be $C(H)=\{g\in G\,|\,  gh=hg\text{ for all }h\in H\}$.
\begin{enumerate}[label=(\alph*)]
	\item Prove that $N(H)$ is a subgroup of $G$.
	\item Prove that $C(H)$ is a normal subgroup of $N(H)$ and that $N(H)/C(H)$ is isomorphic to a subgroup of $\operatorname{Aut}(H)$.
\end{enumerate}

Let $V$ denote the real vector space of polynomials in $x$ of degree at most 3. Let $\mathcal{B}=\{1, x, x^2, x^3\}$ be a basis for $V$ and $T:V\to V$ be the function defined by $T(f(x))=f(x)+f'(x)$.
\begin{enumerate}[label=(\alph*)]
	\item Prove that $T$ is a linear transformation.
	\item Find $[T]_{\mathcal{B}}$, the matrix representation for $T$ in terms of the basis $\mathcal{B}$.
	\item Is $T$ diagonalizable? If yes, find a matrix $A$ so that $A[T]_{\mathcal{B}}A^{-1}$ is diagonal, otherwise explain why $T$ is not diagonalizable.
\end{enumerate}

Let $f(x)=x^3+x+1\in {\bf Z}_5[x]$.
\begin{enumerate}[label=(\alph*)]
	\item Prove that $f(x)$ is irreducible.
	\item Prove that $\langle f(x)\rangle$ is a maximal ideal.
	\item What is the cardinality of ${\bf Z}_5[x]/\langle f(x)\rangle$? Justify.
\end{enumerate}

Let $R$ be a commutative ring. The {\bfseries nilradical} of $R$ is defined to be $N=\{r\in R\,|\, r^n=0\text{ for some }n\in {\bf N}\}$.
\begin{enumerate}[label=(\alph*)]
	\item Prove that $N$ is an ideal of $R$.
	\item Prove that $N$ is contained in the intersection of all prime ideals of $R$.
\end{enumerate}