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\title{MATH0037 Galois Theory 2002, Sheet 1}
\author{Geoff Smith, http://www.bath.ac.uk/$\sim$masgcs}
\date{}
\begin{document}
\maketitle

\begin{enumerate}

\item Algebra and geometry are intimately connected. For example:
{\em Are there three points in the plane $\mathbb R^2$
with integral co-ordinates
which form the vertices of an equilateral triangle?} You can
address this question using one of the rings we have been 
discussing. What about squares and regular pentagons?

\item We have proved a theorem that tells us that a finite 
integral domain is a field. This knowledge makes the following
result fairly easy to prove: {\em Suppose that 
$n \in \mathbb N$, and that $\mathbb Z_n$ denotes the integers
modulo $n$. It follows that $\mathbb Z_n$ is a field if and only if 
$n$ is a prime number.} Do it.

\item Suppose that $R$ is a ring and that $R[X]$ is an integral
domain. Prove that $R$ is an integral domain. 

\item 
\begin{enumerate}
\item[(a)] Which elements of $\mathbb Q[X]$ have a multiplicative
inverse in $\mathbb Q[X]$?
\item[(b)$^*$] Which elements of $\mathbb Q[[X]]$ have a multiplicative
inverse in $\mathbb Q[[X]]$?
\end{enumerate}

\item Suppose that $\alpha \in \mathbb C$. Show that there
is a polynomial $f \in \mathbb R[X]$ with $\mbox{deg} f = 2$
such that $f(\alpha) = 0$ (this notation means that $\alpha$
is substituted for $X$ in $f$).

\item In this question only, $R$ not guaranteed to have 
a commutative law of multiplication, but all the other
ring axioms are satisfied.
\begin{enumerate}
\item[(a)] Suppose that $r^2 = r\ \forall r \in R$. Prove that
$R$ has commutative multiplication.
\item[(b)$^*$] Suppose that $r^3 = r\ \forall r \in R$. Prove that
$R$ has commutative multiplication.
\end{enumerate}
\end{enumerate}
\end{document}
