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\begin{document}

\begin{center}
	\LARGE{Math 31 - Homework 7}
	
	\vspace{2mm}
	\large{Due Friday, August 17}
\end{center}

\paragraph{1.} Let $R$ be a ring, and suppose that $I$ and $J$ are ideals in $R$. Prove that $I \cap J$ is an ideal in $R$.

\paragraph{2.} Let $R$ be a commutative ring, and fix $a \in R$. Define the \textbf{annihilator} of $a$ to be the set
	\[
		\Ann(a) = \left\{ x \in R : xa = 0 \right\}.
	\]
Prove that $\Ann(a)$ is an ideal of $R$.

\paragraph{3.} Let $R$ be a commutative ring. An element $a \in R$ is said to be \textbf{nilpotent} if there is a positive integer $n$ such that $a^n=0$. The set
\[
	\mathop{\text{Nil}}(R) = \left\{ a \in R : a \text{ is nilpotent} \right\}
\]
is called the \textbf{nilradical} of $R$. Prove that the nilradical is an ideal of $R$.

\paragraph{4.} Let $R$ and $S$ be two rings with identity, and let $1_R$ and $1_S$ denote the multiplicative identities of $R$ and $S$, respectively. Let $\varphi : R \to S$ be a nonzero ring homomorphism. (That is, $\varphi$ does not map every element of $R$ to 0.)
\begin{thmenum}
	\item Show that if $\varphi(1_R) \neq 1_S$, then $\varphi(1_R)$ must be a zero divisor in $S$. Conclude that if $S$ is an integral domain, then $\varphi(1_R) 		= 1_S$.
	\item Prove that if $\varphi(1_R) = 1_S$ and $u \in R$ is a unit, then $\varphi(u)$ is a unit in $S$ and
		\[
			\varphi(u^{-1}) = \varphi(u)^{-1}.
		\]
\end{thmenum}

\paragraph{5.} Let $R$ be a commutative ring with identity.
\begin{thmenum}
	\item Fix $\alpha \in R$. Define the \textbf{evaluation homomorphism at $\alpha$} to be the map $\text{ev}_{\alpha} : R[x] \to R$ given by: if $p(x) = a_nx^n + 			\cdots + a_1x+a_0$ is in $R[x]$, then
		\[
			\text{ev}_{\alpha}(p) = a_n\alpha^n + \cdots + a_1\alpha + a_0.
		\]
		Show that $\text{ev}_{\alpha}$ is indeed a ring homomorphism.
		
	\item Determine the kernel of $\text{ev}_{\alpha}$.
	
	\item Suppose now that $R[x]$ is a PID. Show that the kernel of $\text{ev}_\alpha$ is a maximal ideal, and conclude that $R$ must be a field in this case.
\end{thmenum}

\paragraph{6.} Determine whether each of the following polynomials is irreducible over the given field.
\begin{thmenum}
	\item $3x^4+5x^3+50x+15$ over $\Q$.
	\item $x^2+7$ over $\Q$.
	\item $x^2+7$ over $\C$.
\end{thmenum}



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