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\title{Math 421-621\\Spring 2013\\Homework 8}
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\centerline{\it Due Wednesday April 10, 2013.}

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\noindent 1. Let $R$ be a commutative ring with identity. We say that the characteristic of $R$ ($\text{char}(R)$) is the smallest positive integer $n$ such that $nr=0$ for all $r\in R$. If no such positive $n$ exists, we say that $\text{char}(R)=0$. 
\begin{itemize}
\item[a)] (5 pt) Show that if $\mathbb{F}$ is a field, then $\text{char}(\mathbb{F})$ is either $0$ or a positive prime.
\item[b)] (5 pt) Show that if $\mathbb{F}$ is a field, then $\mathbb{F}$ contains $\mathbb{Q}$ or $\mathbb{Z}_p$ for some positive prime $p$.
\item[c)] (5 pt) Show that if $\mathbb{F}$ is a field with finitely many elements, then $\text{char}(\mathbb{F})=p$ and $\vert\mathbb{F}\vert=p^m$ for some positive integer $m$.
\end{itemize}

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\noindent 2. Let $R$ be an integral domain, $\mathbb{F}$ be a field, and $f(x)$ an irreducible polynomial in $\mathbb{F}[x]$.
\begin{itemize}
\item[a)] (5 pt) Show that if $R$ is a PID, then every nonzero prime ideal of $R$ is maximal.
\item[b)] (5 pt) Show that $\mathbb{F}[x]$ is a PID.
\item[c)] (5 pt) Show that $\mathbb{K}:=\mathbb{F}[x]/(f(x))$ is a field containing (an isomorphic copy of) $\mathbb{F}$.
\item[d)] (5 pt) Show that any element of $\mathbb{K}$ is a root of a polynomial with coefficients in $\mathbb{F}$.
\end{itemize}


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