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\title{MATH0037 Galois Theory 2002, Sheet 3}
\author{Geoff Smith, http://www.bath.ac.uk/$\sim$masgcs}
\date{}
\begin{document}
\maketitle
\begin{enumerate}
\item Suppose that $R$ is an integral domain. Let $S = R - \{0\}$.
Define a relation $\sim$ on the set $R \times S$ by
$(r_1, s_1) \sim (r_2, s_2)$ if and only if $r_1s_2 - r_2s_1 = 0$.
\begin{enumerate}
\item Show that $\sim$ is an equivalence relation.
\item Let the equivalence class of $(r,s)$ be denoted ``$r/s$''.
Let the set of equivalence classes be denoted $F_R$.
We attempt to define addition and multiplication on 
$F_R$ by $r_1/s_1 + r_2/s_2 = (r_1s_2 + r_2s_1)/s_1s_2$.
Show that these operations are well-defined.
\item What is the additive inverse of $r/s$ and what is the
multiplicative inverse of $r/s$ if $r \not = 0$?
\item Prove that $F_R$ is a field, and that there is a ring
monomorphism $\alpha: R \rightarrow F_R$ defined by $r \mapsto r/1$
for every $r \in R$.
\item Deduce that every integral domain can be regarded as 
being a subring of a field.
\item In the procedure outlined in this question, what is the field
$F_R$ if $R$ is a field?
\item What is the field $F_{\mathbb G}$ when $\mathbb G$ is the set
of Gaussian integers?
\end{enumerate}
{\em The first four parts of this question form the
bulk of primary school mathematics education.}
\item Suppose that $R$ is a ring and that $S$ is a subring of $R$.
Suppose that $I \unlhd R$. 
\begin{enumerate}
\item Show that $S + I = \{s + \lambda \mid s \in S, \lambda \in I\}$ is
a subring of $R$ and that $I \unlhd (S + I)$.
\item Define a map $\psi: S \rightarrow (S + I)/I$ by $s \mapsto s + I$
for all $s \in S$. Show that $\psi$ is an epimorphism of rings.
\item Deduce that $S \cap I$ is an ideal of $S$, and that
the rings $S/(S \cap I)$, $(S+I)/I$ are isomorphic via a natural
map. What is that map?
\end{enumerate}
\item Construct finite fields of orders $2,3,4,5,7,8,9,11,16$ and $27$.
In each case verify that the non-zero elements of the field you have
built form a {\em cyclic} group under multiplication. (The fact that they
form a group is true in any field, the issue is the cyclicity of that group.)
\end{enumerate}
\end{document}
