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\begin{document}

\noindent{\bf MATHEMATICS} {\bf 362} \hspace{2in} {\bf Name}
\hrulefill

\noindent{\bf Test 3}

\noindent April 9, 2001 \hfill 3 questions

\next
\parindent=0mm


\vspace{4mm}
 {\bf 1}. ({\sl 30 points})

\vspace{3mm}
\noindent Let $L:K$ be a field extension which is finite, normal
and separable.

\vspace{3mm}
(a) By considering the non-identity elements, show that a group
of finite order
$n$ has at most
${n-1 \choose m-1}$ subgroups of
order $m$.

\vspace{2in}

 (b) Prove that there are only finitely many intermediate fields
$M$ in the extension $L:K$. [{\sl Name any theorem you use in
your argument.}]

\vspace{2in}

 (c) If $[L:K]=4$, find the best upper bound you can for the
number of intermediate fields in the extension $L:K$. [{\sl Name any
theorem you use in
your argument.}]



\newpage

{\bf 2}. ({\sl 30 points})

\vspace{3mm}
\noindent  Let $L:K$ be a field extension which is finite, normal
and separable. Let $G$ denote the Galois group $\Gamma(L:K)$.
Suppose that $M$ is an intermediate field, and $M^*$ is the
corresponding subgroup of $G$.

\vspace{3mm}

(a) Let $\phi \in G$, so that $\phi(M)$ is another intermediate
field, which may or may not be equal to $M$. (It would be equal
to $M$ if
$M:K$ is normal.) Prove that
$\phi(M)^* = \phi M^* \phi^{-1}$.

\vspace{2.6in}

(b) A theorem proved in class showed that the following two
statements are equivalent.

\vspace{2mm}
\begin{quote}
(i) $M:K$ is a normal extension.
\end{quote}\begin{quote}
(ii) $M^*$ is a normal subgroup of $G$.
\end{quote}
\vspace{2mm}
Give the proof of the part of the theorem that says that (i)
implies (ii).

\vspace{2.2in}

(c) If $M:K$ is a normal extension, how are $M^*$, $G$ and
$\Gamma(M:K)$ related? [{\sl No proof needed.}]

\newpage

{\bf 3}. ({\sl 40 points})
\vspace{4mm}

Let $L$ be the splitting field of $t^n-1$ over a field $K$, and
let $U$ be the set of all roots of $t^n-1$ in $L$.

\vspace{4mm}
(a) Show that $U$ is a subgroup of the multiplicative group of
the nonzero elements of $L$.

\vspace{1.5in}

(b) State carefully a theorem of finite group theory which can
be used to show that $U$ is cyclic. [{\sl No proof needed.}]

\vspace{1.5in}

(c) Give the proof that the Galois group $\Gamma(L:K)$ is
abelian.
\vspace{2.2in}

(d) What can be said about the Galois group $G$ of a polynomial
of form $t^n-b$ over a field of characteristic $0$? [{\sl No
proof needed.}]



\end{document}


