\documentclass[12pt]{amsart}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage[all]{xy}
\voffset=-1in
\hoffset=-1in
\textheight=10in
\textwidth=6.5in
\begin{document}
\author{}
\title{Math 421-621\\Spring 2013\\Homework 11}
\date{}
\maketitle
\thispagestyle{empty}
\pagestyle{empty}



\centerline{\it Due Wednesday May 1, 2013.}

\centerline{}

\noindent 1. (5 pt) Let $F$ be a field and let $\overline{F}$ be its algebraic closure. Show that $\overline{F}$ is algebraically closed.

\centerline{}

\noindent 2. (5 pt) Show that any field has an algebraic closure.

\centerline{}

\noindent 3. Let $F$ be a field and $f(x)\in F[x]$ be an irreducible polynomial and $K$ the splitting field of $f(x)$ over $F$.
\begin{itemize}
\item[a)] (5 pt) Show that $\alpha$ is a multiple root in $K$ if and only if $\alpha$ is a root of $f^\prime(x)$.
\item[b)] (5 pt) If $\text{char}(F)=0$ then show that $f(x)$ is a separable polynomial.
\item[c)] (5 pt) Show that if $F$ is a finite field, then $f(x)$ is separable.
\item[d)] (5 pt) Show that for all $n$, the polynomial $x^{p^n}-x$ is separable over $\mathbb{Z}_p$.
\item[e)] (5 pt) Show that the set of roots of $x^{p^n}-x$ forms a field extension of $\mathbb{Z}_p$.
\item[f)] (5 pt) Show that for all $n$ there is a unique (up to isomorphism) field of $p^n$ elements.
\end{itemize}


\end{document}