\documentclass{article}
\pagestyle{plain}
\usepackage{amssymb}
\usepackage{latexsym}

\title{MATH0037 Galois Theory 2000}
\author{GCS -- Sheet 4}
\date{}
\begin{document}
\maketitle

\begin{enumerate}
\item Let $R$ be a principal ideal domain which is not
a field. Prove that $x \in R$ is irreducible
if and only if $(x)$ is a maximal ideal {\em (apparently I omitted one of the
two implications in lectures, and now is a good opportunity to fill this
lacuna).}
\item Suppose that $f \in {\mathbb Z}[X]$ and that $f$ is not the zero polynomial. We define the {\em content} $c(f)$ of $f$ to be the greatest common
divisor of the coefficients of $f$. We say that $f$ is {\em primitive}
if $c(f) = 1$.
\begin{enumerate}
\item Prove that if $f, g \in {\mathbb Z}[X]$ are primitive, then $fg$ is primitive. {\em Hint: If $fg$ is not primitive, then there is a prime number
$p$ which divides each of its coefficients.}
\item Show that if $f \in {\mathbb Z}[X]$ and $f$ is not the zero polynomial,
then $f = c(f) \cdot \hat f$ where $\hat f \in {\mathbb Z}[X]$ is a primitive
polynomial.
\item Show that if $f, g \in {\mathbb Z}[X]$, then $c(fg) = c(f) c(g)$.
\item Suppose that $f, g, h \in {\mathbb Q}[X]$ with $0 \not = f
\in {\mathbb Z}[X].$ Suppose also that $f = gh$. Show that there are
rational numbers $\alpha, \beta$ such that $\alpha\beta = 1$ and 
$\alpha f, \beta g \in {\mathbb Z}[X]$. What does this say 
for the factorization of $f$?
{\em Hint: Choose non-zero integers $m,n$ such that $g_1 = mg \in {\mathbb Z}[X]$ and $h_1 = nh \in {\mathbb Z}[X]$. Now $mnf = g_1 h_1.$ Get
very interested in the content of each side.}
\end{enumerate}
\end{enumerate}
\end{document}