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\title{MATH0037 Galois Theory 2002, Sheet 4}
\author{Geoff Smith, http://www.bath.ac.uk/$\sim$masgcs}
\date{}
\begin{document}
\maketitle
\begin{enumerate}
\item Factorize the following polynomials:
$X^2 + 1$,\ $2X^2 - 1$,\ $X^5-1$
\begin{enumerate}
\item in $\mathbb C[X]$,
\item in $\mathbb R[X]$,
\item in $\mathbb Q[X]$.
\end{enumerate}
\item \begin{enumerate} 
\item Consider the ring $\mathbb G$ of Gaussian integers.
Show that $z \in G$ has the property that $z^{-1} \in \mathbb G$ if and only
if $z \overline z = 1$. Determine which Gaussian integers have this property.
\item Let $\omega = e^{2\pi i/3} \in \mathbb C$, 
Consider the ring $R = \{ a + b \omega \mid a, b \in \mathbb Z\}$
(a subring of the complex numbers). Suppose that $r  = x + \omega y$ with
$x, \in \mathbb Z$. Show that 
that $r^{-1} \in R$ if and only if $x^2 + xy + y^2 = 1$. Determine
which elements of $R$ have this property.
\item[(c)$^*$] Consider the ring $S = \{ a + b \sqrt 2 \mid a, b \in \mathbb Z\}$.
Determine which elements $s$ of $S$ have the property that $s^{-1} \in S$.
\end{enumerate}
\item 
\begin{enumerate}
\item Suppose that $F$ is a subring of the ring $R$, and that $F$ is a field.
Show that addition and multiplication in $R$ render
$R$ into a vector space over $F$.
\item Suppose that $R$ is a finite field. Let $S_0 = \{ 0 \} \subseteq R$.
For each integer $i>0$ let $S_{i+1} = S_i \cup \{ x + 1_R \mid  x \in S_i\}.$
Let \[S = \cup_{i \geq 0} S_i.\] Prove that 
$-1_R \in S$ and that $S$ is a subfield of $R$. 
Show that $S$ is isomorphic to $\mathbb Z_p$ for some prime number $p$. 
\item Deduce that if $R$ is a finite field then $|R| = p^n$ for some
integer $n$. {\em Hint: in the notation of part (b), let
$n = \mbox{dim}_S R$.}
\end{enumerate}
\end{enumerate}
\end{document}
