Template problems in ring theory

\begin{enumerate}
	\item Suppose $I$ and $J$ are ideals in a commutative ring $R$ such that $R=I+J$. Prove that the map $f:R\to R/I\times R/J$ given by $f(x)=(x+I,x+J)$ induces the isomorphism
	\[
		R/IJ\cong R/I\times R/J.
	\]
	\item Prove that $\left({\bf Z}/3{\bf Z}\right)[x]/(x^3-x^2-1)\cong \left({\bf Z}/3{\bf Z}\right)[x]/(x^3+x+1)$. ({\itshape Hint:} Use part (a).)
\end{enumerate}

Let $\mathcal{C}([0,1])$ be the (commutative) ring of continuous, real-valued functions on the unit interval, and let
\[
	M=\left\{f\in \mathcal{C}([0,1])\,\mid\, f\left(\frac{1}{2}\right)=0\right\}.
\]
Prove that $M$ is a maximal ideal.

Let $z\in {\bf C}$ be a complex number and let $\epsilon_z:{\bf R}[x]\to {\bf C}$ be the evaluation homomorphism given by $\epsilon_z(p)=p(z)$ for each $p\in {\bf R}[x]$.

\medskip
\begin{enumerate}[label=(\alph*)]
	\item Show that $\ker(\epsilon_z)$ is a prime ideal.
	\item Compute $\ker(\epsilon_{1+i}), \operatorname{im}(\epsilon_{1+i})$ and then state the conclusion of the First Isomorphism Theorem applied to the homomorphism $\epsilon_{1+i}$.
\end{enumerate}

Let ${\bf Z}\left[\sqrt{2}\right]=\left\{a+b\sqrt{2}\,|\, a,b\in{ \bf Z}\right\}$, and let $R\subset \operatorname{M}_2({\bf Z})$ be the ring of all $2\times 2$ matrices of the form $\begin{bmatrix} a & b \\ 2b & a\end{bmatrix}$. Prove ${\bf Z}\left[\sqrt{2}\right]$ is isomorphic to $R$.

\medskip
\noindent {\itshape Hint:} Start by showing the map $a+b\sqrt{2}\mapsto \begin{bmatrix} a & b \\ 2b & a\end{bmatrix}$ is a ring homomorphism.

Let $f(x)=x^3+x+1\in {\bf Z}_5[x]$.
\begin{enumerate}[label=(\alph*)]
	\item Prove that $f(x)$ is irreducible.
	\item Prove that $\langle f(x)\rangle$ is a maximal ideal.
	\item What is the cardinality of ${\bf Z}_5[x]/\langle f(x)\rangle$? Justify.
\end{enumerate}

\begin{enumerate}[topsep=0.1in]
	\item Write down an irreducible cubic polynomial over ${\bf F}_2$.
	\item Construct a field with exactly eight elements and write down its multiplication table.
\end{enumerate}

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}

Let ${\bf Z}[i]=\{a+bi\,\mid \, a,b\in {\bf Z},\, i^2=-1\}$, a subring of ${\bf C}$. Prove there is no ring homomorphism ${\bf Z}[i]\to {\bf Z}_{19}$, but there is a ring homomorphism ${\bf Z}[i]\to {\bf Z}_{13}$. Note a ring homomorphism of commutative rings with $1$ must send $1$ to $1$.

\medskip
\noindent {\itshape Hint:} The group of units in ${\bf Z}_{19}$ is the cyclic group $U(19)$ of order 18, and the group of units in ${\bf Z}_{13}$ is the cyclic group $U(13)$ of order 12.

Let ${\bf Z}_n$ be the ring of integers $\pmod{n}$. There is a ring homomorphism
	\begin{align*}
		{\bf Z}_{28}&\to {\bf Z}_4\times {\bf Z}_7\\
		[m]_{28}&\mapsto ([m]_4,[m]_7)
	\end{align*}
This is an isomorphism by the Chinese Remainder Theorem. Let ${\bf Z}_n^{\times}$ be the group of units of ${\bf Z}_n$. Prove that ${\bf Z}_{28}^{\times}$ is isomorphic to ${\bf Z}_4^{\times}\times {\bf Z}_7^{\times}$.

Let $k\subset K$ be fields, and let $k[X]$ be the polynomial ring in one variable with coefficients in $k$. The {\bfseries evaluation} at $z\in K$  is a ring homomorphism $\varepsilon:k[X]\to K$ defined by $\varepsilon(f(X))=f(z)$. Prove that if $\varepsilon$ is not injective, then $\varepsilon(k[X])$ is a field.

Let $i$ be the imaginary number, let ${\bf Z}[i]=\{a+bi\,\mid \, a,b\in {\bf Z}\}$, a principal ideal domain, and let ${\bf Z}_2$ be the finite ring of integers modulo 2.
	\begin{enumerate}[label=\alph*)]
		\item Define a ring homomorphism from ${\bf Z}[i]\to {\bf Z}_2$. You must prove it is a ring homomorphism.
		\item Find, with proof, a generator for the kernel of your ring homomorphism.
	\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 ${\bf Z}_2=\{0,1\}$ 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 ${\bf F}_9$ denote the field of nine elements.
\begin{enumerate}[label=\alph*)]
	\item Show that each nonzero $a\in {\bf F}_9$ is a root of $X^3-1=(X-1)(X^2+1)(X^4+1)\in {\bf F}_3[X]$.
	\item Use the Pigeonhole Principle to prove that ${\bf F}_9$ has an element of multiplicative order 8. (Include a proof that the Pigeonhole Principle applies.)
\end{enumerate}

Let ${\bf Z}[X]$ be the ring of polynmomials with integer coefficients, and let $K\subset {\bf Z}[X]$ be the kernel of the ``evaluation at $1