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{\bf{\centerline{ Math 362 Exam 2: Tuesday
Mar 30,1999;  12:30--1:45pm}}}
\medskip
{\bf Instructions} Answer questions 1--2.
 You must show all necessary
working to receive full points for a problem.
If you are unable to do one part of a
problem, you may still use the result of
that part in your answer to other parts of
the exam.
\bigskip
\parindent=0pt{\bf{1.}} {\sl (50 points)} 
Let $L:K$ be a finite field extension.
\smallskip

(a) Give a careful definition of the Galois
group $G=\Gamma(L:K)$. \smallskip
(b) For an intermediate field $M$, how is the
corresponding subgroup $M^*$ of the Galois
group defined? Also define the intermediate
field $H^\dagger$ corresponding to a subgroup
$H$ of $G$. Prove carefully that $H\subseteq
(H^\dagger)^*$.\smallskip
(c) Explain carefully what is meant by
saying that the extension $L:K$ is normal
and separable. Give an example of a finite
extension which is not normal.\smallskip
(d) Under what assumptions on the extension
$L:K$ will the Galois correspondence be
bijective. Assuming that it is bijective,
show that if $M$ is an intermediate field and
$\tau\in G$ then
$(\tau(M))^*=\tau M^*\tau^{-1}$.
\smallskip
(e) Assume the Galois correspondence is
bijective. Show that if $M$ and $N$ are
intermediate fields such that $M\cap N=K$,
then $M^*\cup N^*$ generates the group $G$. 
\bigskip

{\bf{2.}} {\sl(50 points)}
Let $\omega$ denote the complex
number $\omega=e^{\pi
i/4}={1\over{\sqrt 2}}(1+i)$ where
$i^2=-1$.\smallskip
 (a) Show that 
$\hbox{Min}(\omega,{\bf Q}, X)=f(X)$
where $f(X)=X^4+1$.
(Hint: show $f$ is irreducible over $\bf Q$
by considering $f(X+1)$). What is the degree
$[{\bf Q}(\omega):{\bf Q}]$?
\smallskip 
(b) Show that the roots of $f$ in $\bf C$  are $\omega^j$ for
$j=1,3,5,7$. Conclude that
${\bf{Q}}(\omega)$ is a splitting field for 
$f$ over $\bf Q$. Is ${\bf Q}(\omega):\bf Q$
 a normal extension? Is it a separable
extension? Is the Galois correspondence
bijective for this extension?
\smallskip
(c) Let $G=\Gamma({\bf Q}(\omega):\bf Q)$ be
the Galois group. Show that $G$ has four
elements and conclude
$G=\{\tau_1,\tau_3,\tau_5,\tau_7\}$ 
where $\tau_j(\omega)=\omega^j$ for each
 $j=1,3,5,7$.\smallskip 

(d) Show that 
$\tau_j^2=\tau_1$ for all $j$ and conclude 
that $G$ is isomorphic to 
${\bf Z}_2\times {\bf Z}_2$.\smallskip

(e) Draw  diagrams of all the
subgroups of $G$ and the corresponding
intermediate fields of 
${\bf Q}(\omega):\bf Q$
illustrating the Galois correspondence.
(As a check on your answer, you should find
five subgroups including $G$ and the trivial
subgroup.)
   
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