Algebra Qual 2017-03

Let $V=\{a_0+a_1\sqrt[3]{2}+a_2\sqrt[3]{4}\mid a_0, a_1, a_2\in {\bf Q}\}\subseteq {\bf R}$. This set is a vector space over ${\bf Q}$.
\begin{enumerate}[label=\alph*)]
	\item Verify $V$ is closed under product (using the usual product operation in ${\bf R}$).
	\item Let $T:V\to V$ be the linear transformation defined by $T(v)=(\sqrt[3]{2}+\sqrt[3]{4}) v$. Find the matrix that represents $T$ with respect to the basis $\{1,\sqrt[3]{2},\sqrt[3]{4}\}$ for $V$.
	\item Determine the characteristic polynomial for $T$.
\end{enumerate}

Suppose $F$ is a field and $A$ is an $n\times n$ matrix over $F$. Suppose further that $A$ possesses distinct eigenvalues $\lambda_1$ and $\lambda_2$ with $\dim \operatorname{Null}(A-\lambda_1 I_n)=n-1$. Prove $A$ is diagonalizable.

\begin{enumerate}[label=\alph*)]
	\item Suppose $N$ is a normal subgroup of a group $G$ and $\pi_N:G\to G/N$ is the usual projection homomorphism, defined by $\pi_N(g)=gN$. Prove that if $\phi:G\to H$ is any homomorphism with $N\leq \ker(\phi)$, then there exists a unique homomorphism $\psi:G/N\to H$ such that $\phi = \psi\circ \pi_N$. (You must explicitly define $\psi$, show it is well defined, show $\phi=\psi\circ\pi_N$, and show that $\psi$ is uniquely determined.)
	\item Prove the:
		\medskip
		{\bfseries Third Isomorphism Theorem.} If $M, N\unlhd G$ with $N\leq M$, then $(G/N)/(M/N)\cong G/M$.
\end{enumerate}

Explicitly list all group homomorphisms $f: {\bf Z}/6{\bf Z} \to {\bf Z}/12{\bf Z}$.

Let $\varepsilon:{\bf R}[x]\to {\bf C}$ be the ring homomorphism that is evaluation at $i$, so $\varepsilon(f)=f(i)$. (Here $i$ denotes the complex number sometimes denoted $\sqrt{-1}$.)
\begin{enumerate}[label=\alph*)]
	\item Prove that $\ker(\varepsilon)=(x^2+1)\subseteq {\bf R}[x]$.
	\item Prove that $(x^2+1)$ is a maximal ideal in ${\bf R}[x]$.
\end{enumerate}