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{\bf \centerline{Math 362 Final  Exam;  Monday, May 5 1997 4:15--6:15pm}}
\noindent{\bf Instructions.} Answer  questions 1--3.
To receive full points, you must justify all your steps by either
explicit argument or by briefly indicating which theorems from class you are
using. On the other hand, you do not have to prove theorems which you quote
unless specifically asked to.  

\bigskip
\noindent{\bf 1.} {\sl (45 points)} Let $K$ be a field of characteristic zero.
\smallskip \item{(a)} Explain  what is meant by saying an extension
field $L$ of $K$ is a radical extension of $K$.
 \smallskip \item{(b)} Define the Galois group of a polynomial $f\in K[t]$. 
\smallskip\item{(c)} Explain what is meant by saying that a
polynomial $f\in K[t]$ is solvable by radicals.
\smallskip\item{(d)} Explain what is meant by saying that a group
$G$ is solvable.
\smallskip\item{(e)} For a normal subgroup $N$ of a  group
$G$, what is the relationship between solvability of $G$, $N$ and
$G/N$? Give an example of a group of order 180 which is    solvable 
and an example of one which is  not solvable. Are there any infinite solvable
groups? \smallskip\item{(f)} Carefully state the theorem giving the relationship
between solvable groups and solvability of equations by radicals.
For which natural numbers  $n$ is every polynomial of degree $n$ over a field of
characteristic zero solvable by radicals?\smallskip 
 \item{(g)} State conditions on a polynomial $f$ of prime degree $p$ over the
rational numbers  which are sufficient to ensure that the Galois group of $f$ is
isomorphic to the symmetric group $S_p$.
\smallskip\item{(h)} Use the result in (g) to show that the extension $L/{\bold
Q}$ has $S_3$ as Galois group, where $L$ is the subfield of ${\bold C}$
 generated
by the roots of $t^3+3t+3$.   \medskip
\noindent{\bf 2.} {\sl (45 points)} Let $K={\bold Q}(\omega)$
where $\omega=e^{2\pi i/9}\in {\bold C}$.\smallskip
\item{(a)} Let $f=t^6+t^3 +1=(t^9-1)/(t^3-1)\in{\bold Q}[t]$. Applying
Eisenstein's criterion to $f(t+1)$, deduce that $f$ is irreducible over ${\bold
Q}$. Conclude that $f$ is the minimal polynomial of $\omega$ over ${\bold
Q}$.\smallskip 
  \smallskip \item{(b)} Show that the  distinct roots of $f$ in $K$
are the elements $\omega^i$ for $i=1$, $2$, $4$, $ 5$, $7$ and $8$, and conclude
that for each such $i$, there is an automorphism $\sigma_i$ of ${\bold Q}(\omega)$
with $\sigma_i(\omega)=\omega^i$.  
  Show that the Galois group $G=\Gamma(K/{\bold Q})$ consists of  these
$6$ automorphisms $\sigma_i$.\smallskip
\item{(c)} Calculate
$(\sigma_i\sigma_j)(\omega)$ and hence show that the map $\sigma_i\mapsto
\overline i$ is a group isomorphism between $G$ and the multiplicative group
${\bold Z}_9^*$ of units  in ${\bold Z}_9$ (here, $\overline i$ denotes the congruence
class of $i$ in ${\bold Z}_9$). \smallskip\item{(d)} Show  that $G$ is a cyclic group
of order $6$ generated by $\sigma_2$. 
\smallskip\item{(e)}  What conditions must a finite field extension satisfy to
ensure that the Galois correspondence is bijective?  Prove that these
conditions are satisfied by $K/{\bold Q}$.\smallskip
\item{(f)} For each of the non-trivial proper subgroups of $G$, find the
corresponding fixed field in $K$, proving carefully that your answer is correct.
Draw  diagrams showing all  subgroups of $G$ and the corresponding intermediate
fields, exhibiting the Galois correspondence.\medskip

\noindent{\bf 3.} {\sl (60 points)}
\smallskip \item{(a)} For which integers $p$ and $q$ does  there exist a finite
field ${\bold {GF}}(q)$ with $q$ elements and characteristic $p$? 
\smallskip\item{(b)} For $p$, $q$ satisfying the conditions you gave in
(a), describe an explicit polynomial in $ {\bold Z}_p[t]$
 such that ${\bold {GF}}(q)$ is the splitting field
(and also the set of roots) of $f$. 
\smallskip\item{(c)} Explain using the results mentioned in  (b)  why every
extension of finite fields is  separable and normal (for separability, you'll
have to compute the formal derivative of the polynomial in (b)).
  \smallskip\item{(d)} For which integers $q'$ does  ${\bold
{GF}}(q)$ have a subfield with $q'$ elements? How many subfields with $q'$
elements are there?
 \smallskip\item{(e)} Let $E$ be a finite field, let $F$ be
the splitting field of an irreducible polynomial $f\in E[t]$ and let $\alpha$ be
a root of $f$ in $F$. Show that $F=E(\alpha)$. (hint: $E(\alpha)/E$ is normal, by
(c)). 
\smallskip\item{(f)}    For which of the
finite fields  ${\bold {GF}}(q)$  is it true that every
non-identity element of the multiplicative group ${\bold {GF}}(q)^*$ is a
generator? Give  an example (with $q\geq 5$, to avoid trivialities) for which
this holds, and an example (with $q\geq 5$) for which it doesn't hold.  
\smallskip
\par\noindent Let $P={\bold Z}_2$ and
$f=t^2+t+1$, $g=t^3+t+1$ and $h=t^5+t^2+1$ in the polynomial ring $P[t]$.
The next parts (h)--(k) of this question will show that the splitting field $F$
of the degree ten polynomial $fgh$ over $P$ is isomorphic to ${\bold
{GF}}(2^{30})$. \smallskip\item{(g)} Draw a diagram showing all the subfields
${\bold {GF}}(q)$ of ${\bold {GF}}(2^{30})$ and the inclusion relations amongst
them. 
   \smallskip\item{(h)}  
Show that $f$ is the unique irreducible quadratic polynomial over $P$, and that
$g$ and $h$  are also irreducible (hint; note that a reducible quintic with no
root must have an irreducible quadratic factor).  \smallskip\item{(i)}  Let
$\alpha$, $\beta$, $\gamma$ be roots in $F$ of $f$, $g$ and $h$, respectively.
Determine $[P(\alpha)\colon P]$, $[P(\beta)\colon P]$ and $[P(\gamma)\colon P]$,
and  show that $n:=[F\colon P]$ is divisible by $30$. \smallskip\item{(j)} Show
that $F$ has a subfield ${\bold {GF}}(2^{30})$ with $2^{30}$ elements. Then show
that $P(\alpha)$, $P(\beta)$ and $P(\gamma)$ are all contained in ${\bold
{GF}}(2^{30})$.\smallskip \item{(k)} Using (e), show that all roots of $fgh$ are
in ${\bold {GF}}(2^{30})$ and conclude that $F={\bold {GF}}(2^{30})$.
\smallskip\item{(l)} Suppose that the non-constant polynomial $f$ over the
finite field $F$ has factorization $f=f_1\ldots f_n$ where the $f_i$ are (not
necessarily distinct) irreducible polynomials over $F$ of degree $n_i$. What
degree $[E:F]$ would you expect for the splitting field $E$ of $f$ over $F$?
Justify your answer if you can.   

  
    


  
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