Algebra Qual 2019-09

Let $P_3$ be the real vector space of all real polynomials of degree three or less. Define $L:P_3\to P_3$ by $L(p(x))=p(x)+p(-x)$.
\begin{enumerate}[label=\alph*)]
	\item Prove $L$ is a linear transformation.
	\item Find a basis for the null space of $L$.
	\item Compute the dimension of the image of $L$.
\end{enumerate}

Let $T:{\bf R}^3\to {\bf R}^3$ be the linear transformation that rotates counterclockwise around the $z$-axis by $\frac{2\pi}{3}$.
\begin{enumerate}[label=\alph*)]
	\item Write the matrix for $T$ with respect to the standard basis $\left\{\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ 1 \\ 0\end{bmatrix},\begin{bmatrix} 0 \\ 0 \\ 1\end{bmatrix}\right\}$.
	\item Write the matrix for $T$ with respect to the basis $\left\{\begin{bmatrix} \frac{\sqrt{3}}{2} \\ -\frac{1}{2} \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ 1 \\ 0\end{bmatrix},\begin{bmatrix} 0 \\ 0 \\ 1\end{bmatrix}\right\}$.
	\item Determine all (complex) eigenvalues of $T$.
	\item Is $T$ diagonalizable over ${\bf C}$? Justify your answer.
\end{enumerate}

Suppose $G$ is a cyclic group of order $n$, and $t\in G$ is a generator.
\begin{enumerate}[label=\alph*)]
	\item Give a positive integer $d$ such that $t^{-1}=t^d$.
	\item Let $c$ be an integer and let $m=\gcd(n,c)$. Prove that the order of $t^c$ is $\frac{n}{m}$.
\end{enumerate}

Suppose $G$ is a group, $H$ and $K$ are normal subgroups of $G$, and $H\leq K$.
\begin{enumerate}[label=\alph*)]
	\item Define a group homomorphism from $K$ to $G/H$.
	\item Compute the kernel of the homomorphism in (a), and apply the First Isomorphism Theorem.
\end{enumerate}

Let $I\subseteq {\bf Z}[x]$ denote the set of all polynomials with even constant term.
\begin{enumerate}[label=\alph*)]
	\item Prove that $I$ is an ideal.
	\item Prove that $I$ is not a {\itshape principal} ideal.
\end{enumerate}