Algebra Qual 2022-01

Let $\sigma:G\to H$ be a group epimorphism. Let $N$ be a normal subgroup of $G$ and $K=\sigma(N)$, the image of $N$ in $H$.
\begin{enumerate}[label=\alph*)]
	\item Prove that $K$ is a normal subgroup of $H$. Give an example to show that this is not true if $\sigma$ is not onto.
	\item Under what conditions does $\sigma$ induce a homomorphism $G/N\to H/K$, and when is this an isomorphism? Prove your answer.
\end{enumerate}

The dihedral group, $D_8$, is the group of eight rigid symmetries of a square. Prove that $D_8$ is not the internal direct product of two of its proper subgroups.

Let $A$ be a commutative ring with $1$. The {\bfseries dimension} of $A$ is the maximum length $d$ of a chain of prime ideals $\mathfrak{p}_0\subsetneq \mathfrak{p}_1\subsetneq \cdots \subsetneq \mathfrak{p}_d$. Prove that if $A$ is a PID, the dimension of $A$ is at most 1.

Let ${\bf Z}_2=\{0,1\}$ be the field of two elements. The quotient ring ${\bf Z}_2[x]/\langle x^3+x+1\rangle$ is a field of cardinality 8, containing ${\bf Z}_2$. Let $\pi:{\bf Z}_2[x]\to {\bf Z}_2[x]/\langle x^3+x+1\rangle$ be the natural projection.
\begin{enumerate}[label=\alph*)]
	\item Write down a set of eight distinct coset representatives for the elements of this field.
	\item Determine the multiplicative inverse of $\pi(x)$ in terms of your coset representatives.
\end{enumerate}

Let $T:{\bf R}^3\to {\bf R}^3$ be the orthogonal projection to a $1$-dimensional linear subspace $L\subset {\bf R}^3$.
\begin{enumerate}[label=\alph*)]
	\item List the eigenvalues of $T$.
	\item Write the characteristic polynomial $p_T(x)$ for $T$.
	\item Is $T$ diagonalizable? Justify your answer.
\end{enumerate}