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{\bf{\centerline{ Math 362 Exam 1; Thursday
Feb 25,1999  12:30--1:45pm}}}
\medskip
{\bf Instructions} Answer questions 1--3. Use
your time wisely; the questions are not of
equal value. 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 (25 points)}
\smallskip

(a) Let $L:K$ be a field extension. Explain
what is meant by the degree $[L:K]$ of the
 extension. If $M$ is an intermediate field,
what is the relationship between $[L:M]$,
$[M:K]$ and $[L:K]$?
\smallskip

(b) Decide which of the field extensions
(i) ${\bf C}:{\bf R}$ and (ii) ${\bf
R}\colon {\bf Q}$ are 
(I) simple, (II) finite or
(III) algebraic.  \smallskip 



\bigskip

{\bf{2.}} {\sl(25 points)}\smallskip
(a) Let $K$ be a field, and $f\in K[t]$ be an
irreducible polynomial over $K$.
Explain how to construct a simple extension
$L=K(\alpha)$ of $K$ in which $f$ has a zero
$\alpha$. What is the relationship between
the degree of the polynomial $f$ and the
degree of the extension $[L:K]$?

\smallskip

(b) Take $K={\bf Z}_2$ and $f=t^3+t+1$ in (a).
Explain why the elements $1,\alpha,\alpha^2$
form a basis of $L$ over $K$.  Express
each of
$\alpha^5$ and
$(\alpha+1)^{-1}$ as a $K$-linear combination
of these basis elements.\bigskip

{\bf 3.} {\sl(50 points)} Consider the field
extension $\bf{Q}(\alpha,\omega):\bf{Q}$ where
$\alpha=\root 3 \of 2$ and $\omega= e^{2\pi
i/3}$.\smallskip

(a) Show that 
$f=\hbox{min}(\omega,{\bf{Q}},X)=X^2+X+1$ and
$g=\hbox{min}(\alpha,{\bf{Q}},X)=X^3-2$. Show
that the roots of $f$ in the
complex numbers are $\omega^j$, $1\leq j\leq
2$   and those of $g$ are $\alpha \omega^i$,
$0\leq i\leq 2$.
\smallskip

(b) Show that
$[{\bf{Q}}(\alpha,\omega):{\bf{Q}}]=6$
(Hint: use (a) to show that the degree is
divisible by 2 and by 3 and is less than or
equal to 6).\smallskip

(c) Suppose that $K(\beta):K$ and
$L(\gamma):L$ are simple algebraic
extensions, and $\theta\colon K\rightarrow L$
is a field isomorphism. Give the necessary and
sufficient condition relating  $\theta$ and
the minimum polynomials of
$\beta$ and
$\gamma$   so that there is an isomorphism
of the extensions $K(\beta):K$ with
$L(\gamma):L$ compatible with
$\theta$ and mapping $\beta$ to
$\gamma$.\smallskip

(d) Use (c) to show that the Galois group of
$\bf{Q}(\alpha,\omega):\bf{Q}$ consists of
six elements $\sigma_{i,j}$ ($0\leq i\leq 2$,
$1\leq j\leq 2$) with
$\sigma_{i,j}(\alpha)=\alpha \omega^i$ and
$\sigma_{i,j}(\omega)=\omega^j$.
\smallskip

(e) Show the Galois group in (d) is not
abelian. Can you identify which familiar group
with
$6$ elements is isomorphic to the Galois
group?
 


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