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\title{MATH0037 Galois Theory 2000}
\author{GCS -- Sheet 3}
\date{}
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\maketitle

{\it All rings are
commutative with 1 as far as this course is concerned}
\begin{enumerate}
\item Suppose that $R$ is a ring and that $S$ is a subring
of $R$. Suppose that $S$ happens to be a field. Show that
$R$ equipped with $+$ and $\times$ can be regarded as
an vector space over $S$.
\item Suppose that $K$ is a subfield of $L$ which in turn 
is a subfield of $M$.  Let the finite sequence $(a_i)$
($1 \leq i \leq m$) be a $K$-basis of $L$ (i.e. an ordered
basis of $L$ regarded as a vector space over $K$).
Let the finite sequence $(b_j)$ 
($1 \leq j \leq n$) be an $L$-basis of $M$.
Show that the $mn$ quantities $a_ib_j$
form a $K$-basis of $M.$
Deduce that 
\[ \hbox{dim}_K M = \hbox{dim}_K  L \ \cdot \ \hbox{dim}_L M.\] 
\item What are the following?
\begin{enumerate}
\item $\hbox{dim}_{\mathbb R} {\mathbb R}?$
\item $\hbox{dim}_{\mathbb C} {\mathbb C}?$
\item $\hbox{dim}_{\mathbb R} {\mathbb C}?$
\item $\hbox{dim}_{\mathbb Q} {\mathbb Q}?$
\item $\hbox{dim}_{\mathbb Q} {\mathbb R}?$
\item $\hbox{dim}_{\mathbb Q} {\mathbb C}?$
\end{enumerate}
\item Let $K$ be the subset of $\mathbb R$ consisting
of numbers of the form $a + b \sqrt 2$ where $a, b \in \mathbb Q$.
\begin{enumerate}
\item Prove that $K$ is a subfield of $\mathbb R$. 
{\it Feel free to skip the obvious parts; the interesting
aspect of this problem is to show that a non-zero element
of $K$ has a multiplicative inverse in $K$.}
\item Calculate $\hbox{dim}_{\mathbb Q} K$.
\item Demonstrate that $\sqrt[3]{2} \not \in K.$
\end{enumerate} 
\item Given a rectangle in the Euclidean plane, show how
to to construct a square of the same area using only an
infinitely fine pencil and an arbitrarily long
unmarked straight edge.
\end{enumerate}
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