Algebra Qual 2016-03

Without using Cauchy's Theorem or the Sylow theorems, prove that every group of order 21 contains an element of order three.

Suppose $G$ is a group that contains normal subgroups $H,K\unlhd G$ with $H\cap K=\{e\}$ and $HK=G$. Prove that $G\cong H\times K$.

Let $R$ be a commutative ring.

\medskip
\begin{enumerate}[label=(\alph*)]
	\item Prove that the set $N$ of all nilpotent elements of $R$ is an ideal.
	\item Prove that $R/N$ is a ring with no nonzero nilpotent elements.
	\item Show that $N$ is contained in every prime ideal of $R$.
\end{enumerate}

Let $z\in {\bf C}$ be a complex number and let $\epsilon_z:{\bf R}[x]\to {\bf C}$ be the evaluation homomorphism given by $\epsilon_z(p)=p(z)$ for each $p\in {\bf R}[x]$.

\medskip
\begin{enumerate}[label=(\alph*)]
	\item Show that $\ker(\epsilon_z)$ is a prime ideal.
	\item Compute $\ker(\epsilon_{1+i}), \operatorname{im}(\epsilon_{1+i})$ and then state the conclusion of the First Isomorphism Theorem applied to the homomorphism $\epsilon_{1+i}$.
\end{enumerate}

Let $T:{\bf R}^3\to {\bf R}^3$ be the linear transformation that expands radially by a factor of three around the line parameterized by $L(t)=\begin{bmatrix} 2 \\ 2 \\ -1\end{bmatrix} t$, leaving the line itself fixed (viewed as a subspace).

\medskip
\begin{enumerate}[label=(\alph*)]
	\item Find an eigenbasis for $T$ and provide the matrix representation of $T$ with respect to that basis.
	\item Provide the matrix representation of $T$ with respect to the standard basis.
\end{enumerate}