Algebra Qual 2020-01

Let $A$ be a real $n\times n$ matrix and let $A^{\top}$ denote its transpose.
\begin{enumerate}[label=\alph*)]
	\item Prove that $(A{\bf v})\cdot {\bf w}= {\bf v}\cdot (A^{\top}{\bf w})$ for all vectors ${\bf v},{\bf w}\in {\bf R}^n$. {\itshape Hint:} Recall that the dot product ${\bf u}\cdot {\bf v}$ equals the matrix product ${\bf u}^{\top}{\bf v}$.
	\item Suppose now $A$ is also symmetric, i.e., that $A^{\top} = A$. Also suppose ${\bf v}$ and ${\bf w}$ are eigenvectors of $A$ with different eigenvalues. Prove that ${\bf v}$ and ${\bf w}$ are orthogonal.
\end{enumerate}

Let $M_4({\bf R})$ denote the 16-dimensional real vector space of $4\times 4$ matrices with real entries, in which the vectors are represented as matrices. Let $T:M_4({\bf R})\to M_4({\bf R})$ be the linear transformation defined by $T(A)=A-A^{\top}$.
\begin{enumerate}[label=\alph*)]
	\item Determine the dimension of $\operatorname{ker}(T)$.
	\item Determine the dimension of $\operatorname{im}(T)$.
\end{enumerate}

Let $G$ be a group. For each $a\in G$, let $\gamma_a$ denote the automorphism of $G$ defined by $\gamma_a(b)=aba^{-1}$ for all $b\in G$. The set $\operatorname{Inn}(G)=\{\gamma_a:a\in G\}$ is a subgroup of the automorphism group of $G$, called the subgroup of {\bfseries inner automorphisms}.

\medskip
Prove that $\operatorname{Inn}(G)$ is isomorphic to $G/Z(G)$, where $Z(G)$ is the center of $G$.

Let ${\bf Z}_n$ denote the cyclic group of order $n$. Suppose $m\in {\bf N}$ is relatively prime to $n$. Define the function $\mu_m:{\bf Z}_n\to {\bf Z}_n$ by $m[a]_n=[ma]_n$.
\begin{enumerate}[label=\alph*)]
	\item Prove that the map $\mu_m$ is a well-defined automorphism of ${\bf Z}_n$.
	\item Prove that any automorphism of ${\bf Z}_n$ has the form $\mu_m$ for some $m$.
\end{enumerate}

Let $F$ be a field and $F[x]$ be the polynomial ring, which is a principal ideal domain. Let $R=\{f\in F[x]:f'\in (x)\}$, where $(x)\subset F[x]$ is the ideal generated by $x$, and $f'$ is the (formal) derivative of the polynomial $f$. It is a fact that $R$ is a subring of $F[x]$.
\begin{enumerate}[label=\alph*)]
	\item Prove that $x^2$ and $x^3$ are irreducible elements of $R$.
	\item Let $(x^2,x^3)$ be the ideal in $R$ generated by $x^2$ and $x^3$. Prove this is a proper ideal of $R$.
	\item Prove that $(x^2,x^3)$ is not a {\itshape principal} ideal of $R$.
\end{enumerate}