Algebra Qual 2021-09

Let $n$ be a number between $0$ and $10$. Compute $n^{111}\pmod{11}$, expressing your answer as a number between $0$ and $10$. Give as detailed a proof as you can, justifying every step, no matter who trivial you think it is.

Let $G$ be a group of order $2n$ for some positive integer $n > 1$.
\begin{enumerate}[label=\alph*)]
	\item Prove there exists a subgroup $K$ of $G$ of order $2$.
	\item Suppose $K$ in (a) is a \underline{normal} subgroup. Prove that $K$ is contained in the center $\operatorname{Z}(G)$. (Recall $\operatorname{Z}(G)=\{a\in G\mid ab=ba\text{ for all }b\in G\}$.)
\end{enumerate}

Consider the additive group of integers ${\bf Z}$.
\begin{enumerate}[label=\alph*)]
	\item Prove that every subgroup of ${\bf Z}$ is a cyclic group.
	\item Prove that every homomorphic image of ${\bf Z}$ is a cyclic group.
	\item Now consider the {\itshape ring} ${\bf Z}$. Exhibit a prime ideal of ${\bf Z}$ that is not maximal.
\end{enumerate}

Let $i\in {\bf C}$ be the usual root of unity, with $i^2=-1$, and let ${\bf Z}[i]=\{a+bi\mid a,b\in {\bf Z}\}$ be the ring of Gaussian integers.
\begin{enumerate}[label=\alph*)]
	\item Prove that there exists a (nonzero) ring homomorphism ${\bf Z}[i]\to {\bf Z}_5$.
	\item Compute the kernel of your homomorphism explicitly, and state the conclusion given by the First Isomorphism Theorem. ({\itshape Hint:} The kernel requires two generators.)
\end{enumerate}

Let $a$ and $b$ be real numbers and let $A\in {\bf R}^{3\times 3}$ with each diagonal entry equal to $a$ and each off-diagonal entry equal to $b$.
\begin{enumerate}[label=\alph*)]
	\item Determine all eigenvalues and representative eigenvectors of $A$ together with their algebraic multiplicities.
	
	\medskip
	\noindent ({\itshape Hint:} $A=(a-b)I+bJ$ where $J$ is the $3\times 3$ matrix each of whose entries equals $1$.)
	
	\item Is $A$ diagonalizable? Justify your answer.
	\item Determine the minimal polynomial of $A$.
\end{enumerate}