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{\bf{\centerline{ Math 362 Final Exam:
Wednesday May 5,1999;  2:00--4:00pm}}}
\medskip
{\bf Instructions} Answer questions 1--4.
 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 (30 points)} 
Let $L:K$ be a  field extension.
\smallskip

(a) Explain carefully what is meant by saying
that this extension is (i) finite (ii)
algebraic (iii) simple  (iv) normal or (v)
separable.
\smallskip (b) For each of these
properties, give an example of a
field extension $L:K$ with $L\neq K$ which
has this property, and an example of a field
extension which doesn't have this property
(in this part of the exam only, you are NOT
required to prove that the extensions you
give do or do not have the
property).\smallskip

{\bf 2.} {\sl(40 points)} Let $K$ be
a field of characteristic zero. \smallskip (a)
Explain what is meant by a solvable group.
Give an example of a solvable group of order
240 and an example of a group of order 240
which is not solvable.
\smallskip (b) Explain what is meant by
saying that an extension  $L:K$ of
$K$ is radical.
\smallskip (c) Define the Galois group of a
polynomial
$f\in K[t]$. 
Explain what is meant by saying that $f$
 is solvable by radicals.
\smallskip (d) Carefully state the theorem
giving necessary and sufficient conditions
for $f$ to be solvable by
radicals.
\smallskip (e) State conditions on a
polynomial $f$ of prime degree $p$ over the
rational numbers which are sufficient to
ensure  it has the symmetric group
$S_p$ as its Galois group.
\smallskip (f) Use the theorems in parts (d)
and (e) to show that the polynomial $t^5-4t-2$
over
$\bf Q$ is not solvable by radicals.   
\bigskip

{\bf{3.}} {\sl(40 points)}
Let $\omega$ denote the complex
number $\omega=e^{2\pi
i/10}$, so $\omega^{10}=1$.\smallskip
(a) Let $f(X)=X^4-X^3+X^2-X+1$. Show that
$f(X)(X+1)(X^5-1)=X^{10}-1$.
Conclude that the four roots of $f$ in
the complex numbers are
$\omega^i$ for $i=1,3,7,9$.\smallskip  
 (b) Show that 
$\hbox{Min}(\omega,{\bf Q}, X)=f(X)$.
(Hint: show $f$ is irreducible over $\bf
{Q}$ by   considering
$f(X-1)$).
\smallskip 
(c) State necessary and sufficient conditions
on a  field extension $L:K$ of finite degree
for the Galois correspondence to be bijective.
Prove that these conditions are satisfied by
the extension $\bf{Q}(\omega):\bf{Q}$.

\smallskip
(d) Let $G=\Gamma({\bf Q}(\omega):\bf Q)$ be
the Galois group. Show that 
$G=\{\tau_1,\tau_3,\tau_7,\tau_9\}$ 
where $\tau_j(\omega)=\omega^j$ for 
 $j=1,3,7,9$, and that
$\tau_i\tau_j=\tau_k$ where $k\equiv
ij \pmod {10}$. Conclude that $G$ is a cyclic
group.
\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.
\vfill\eject
{\bf 4.} {\sl (40 points)} Let $K={\bf Z}_2$
and $f=t^4+t+1\in K[t]$. 
Let $M$ be a splitting field of $f$ over $K$
and $L=K(\alpha)$ where $\alpha$ is a root of
$f$ in $M$.\smallskip


(a) Show that $f$ is irreducible over $k$ and
conclude that $1,\alpha,\alpha^2,\alpha^3$ is
a $K$-basis of $L$. How many elements does
$L$ have? For which integers $q$ does $L$
have a subfield with $q$ elements?
\smallskip
(b) Explain why $M=L$.
\smallskip
(c) Show that the element $u=\alpha+1$ is a
generator of the multiplicative group of $L$.
\smallskip (d) Let $\theta:L\rightarrow L$
be the Frobenius automorphism of $L$, given
by $\theta(x)=x^2$ for $x\in L$. Show that
the fixed field of $\theta^2$ is
$\{0,1,u^5,u^{10}\}$. What is
$\hbox{Min}(u^{10},K,X)$? 
\smallskip 

(e) For a positive integer $n$, describe the
factorization into irreducible polynomials
over $K$ of the polynomial
$t^{2^n}-t$.  Determine the number of irreducible
polynomials of degree $12$ over $K$.

   
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