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\title{MATH0037 Galois Theory 2000}
\author{GCS -- Sheet 7}
\date{}
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\maketitle
\begin{enumerate}
\item Suppose that $\alpha \in \mathbb C$ and that $\mathbb Q[\alpha]$
is a field. Prove that $\alpha$ is algebraic.
\item Suppose that $L$ is a field and that $\alpha: L \rightarrow L$
is an automorphism of $L$. Let 
\[ \hbox{Fix}(\alpha) = \{ \lambda \in L \mid (\lambda) \alpha = \lambda\}.\]
Show that $\hbox{Fix}(\alpha)$ is a subfield of $L$.
\item Suppose that $h \in \mathbb C[X]$. We define $h'$ to be the
polynomial which is the ``derivative of $h$ with respect to $X$''.
Now suppose that $K$ is a subfield of the complex numbers
and that $f \in K[X]$ is irreducible as an element of $K[X]$.
\begin{enumerate}
\item Prove that $(f) + (f') = K[X]$. 
\item Deduce that that there is no complex number $\alpha$
which is simultaneously a root of $f$ and of $f'$.
\item Deduce that there is no complex number $\beta$ such that
$f = [X - \beta]^2 r$ for some $r \in K[X]$.
\end{enumerate}
\item Show that for each odd natural
number $n$ there is a field extension
$\mathbb Q \leq K$ where 
$|K:\mathbb Q| = n$ and $\hbox{Gal}_{\mathbb Q}K = 1.$ 
\end{enumerate}
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