\documentclass{article}
\usepackage{amsfonts,amsmath}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\usepackage{enumerate}
\begin{document}
\section*{Math 403/503 Spring 2024}
In this class, by
\begin{itemize}
\item a \emph{ring} $R$ we mean a \emph{ring with identity $1_R$};
\item a \emph{subring} $S\subseteq R$ we mean \emph{subring containing the identity $1_R$};
\item a \emph{ring homomorphism} $\varphi:R\to T$ we mean \emph{ring homomorphism sending $1_R$ to $1_T$}.
\end{itemize}
\subsection*{Homework 6, due March 20}
\begin{enumerate}
\item Let $R$ be a commutative ring. The \emph{radical} of an ideal $I\subseteq R$, denoted $\sqrt{I}$, is defined by
\[\sqrt{I}=\{a\in R\mid a^n\in I\text{ for some integer $n>0$}\}.\]
Show that $\sqrt{I}$ is an ideal of $R$.
\item Let $R$ be a commutative ring. The \emph{nilradical} of $R$ is defined as $\mathcal{N}(R)=\sqrt{(0)}$. In other words, $\mathcal{N}(R)$ is the set of all nilpotent elements of $R$:
\[\mathcal{N}(R)=\{a\in R\mid a^n=0\text{ for some integer $n>0$}\}.\]
Show that $\mathcal{N}(R)$ is equal to the intersection of all prime ideals of $R$. (\emph{Hint:} If $a\in R$ is not nilpotent, the set of all ideals not intersecting $\{a^n\mid n\ge 0\}$ has a maximal element; show it is a prime ideal.)
\item Let $R$ be a commutative ring. Show that $R$ is a field if and only if $R$ has exactly two ideals ($\{0\}$ and $R$ itself). (This shows that fields are precisely the commutative simple rings.)
\item Prove the Third Isomorphism Theorem for rings: Let $R$ be a ring and $I$ and $J$ be ideals of $R$, where $J\subseteq I$. Then
\[R/I \cong \frac{R/J}{I/J}.\]
\item Show that if $R$ is an integral domain, then $R[x]$ is an integral domain. Conclude that if $F$ is a field, then the ring $F[x_1,x_2,\ldots,x_n]$ of polynomials in $n$ variables is an integral domain.
\item Show that $x^p-x\in\Z_p[x]$ has $p$ distinct zeros in $Z_p$, for any prime $p$. Conclude that
\[x^p-x = x(x-1)(x-2)\cdots(x-(p-1)).\]

\item Which of the following polynomials in $\Q[x]$ are irreducible?
\begin{enumerate}[{\rm (a)}]
\item $x^4 -2x^3 +2x^2 +x+4$
\item $3x^5 -4x^3 -6x^2 +6$
\item $x^4 -5x^3 +3x-2$
\item $5x^5 -6x^4 -3x^2 +9x-15$
\end{enumerate}
\end{enumerate}
\end{document}
