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{\bf \centerline{Math 362  Exam 1;  Mon, Feb24, 1997 12:50--1:40pm}}



 \bigskip
\noindent{\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 later parts
of the exam. Calculators may be used if desired.
\bigskip
\noindent{\bf 1.} {\sl (30 points)}\smallskip
\item{(a)} Let $L:K$ be a field extension. Explain carefully what is meant
 by the degree $[L:K]$ of the extension. If $M$ is an intermediate field, what
is the relation between $[L:M]$, $[M:K]$ and $[L:K]$?\smallskip
\item{(b)} Explain what is meant by saying that a field extension
$L:K$ is  (i) simple, (ii) finite  or  (iii) algebraic.\smallskip
\item{(c)} Show that ${\bold Q}(\sqrt 2,\sqrt 7)$ is a simple 
 extension of ${\bold Q}$.\smallskip
\item{(d)} Let $F=K(t)$ be the field of rational functions
in an indeterminate $t$
 over a field $K$, and
$\alpha=t^3+(2+t)^{-1}\in F$. Set
$L=K(\alpha)$ (a subfield of $F$). Show that $t$ is algebraic over $L$.
\smallskip  \bigskip
  
\noindent{\bf 2.}  {\sl(25 points)}\smallskip
\item{(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 $\alpha$. How is irreducibility used in the proof that your $L$
is a field? What is the relation between the degree of the polynomial $f$ and the
extension degree $[L:K]$?\smallskip \item{(b)} Taking $K={\bold Z}_3$ in (a), 
find a suitable polynomial $f$ so the resulting field $L$ contains $9$ elements.
Express each of
 $\alpha^3$ and
$(\alpha+1)^{-1}$ in the form $c_0+c_1\alpha$ with the $c_i\in {\bold Z}_3$.
 Are these expressions unique? 


 \bigskip\noindent{\bf 3.} {\sl(45 points)} Let
 $\alpha=\root 5 \of 2\in {\bold
R}$ and $\omega=e^{2\pi i/5}\in {\bold C}$. 
Define polynomials $f$ and $g$ in ${\bold Q}[X]$ by
 $g=X^4+X^3+X^2+X+1$ and $f=X^5-2$.\smallskip

\item{(a)}  
State Eisenstein's criterion and use it show $g(X+1)$ (and hence $g$) is
irreducible over ${\bold Q}$. Prove that that $g=\hbox{\rm min}(\omega, {\bold
Q}, X)$ and conclude that  $[{\bold Q}(\omega):{\bold Q}]=4$.\smallskip 
\item{(b)}Show similarly
that 
 $f=\hbox{\rm min}(\alpha, {\bold Q},X)$ and
compute $[{\bold Q}(\alpha):{\bold Q}]$. Conclude that
$[{\bold Q}(\alpha,\omega):{\bold
Q}(\omega)]\leq 5$. (Hint; $\hbox{\rm min}(\alpha, {\bold
Q}(\omega), X)$ divides $f$ in ${\bold Q}(\omega)[X]$).  \smallskip
\item{(c)} Prove that $[{\bold Q}(\alpha,\omega):{\bold Q}]$=20. (Hint; show the
degree is divisible by $4$ and by $5$, and is less than or equal to $20$.) 
\smallskip 
\item{(d)} Explain what is meant by an isomorphism from a 
 field extension
$K:M$ to a field extension  $L:N$.
Show that any automorphism of ${\bold Q}(\omega):{\bold Q}$
(i.e. an isomorphism of this extension with itself)
must map $\omega$  to a root $\omega^j$, $1\leq j\leq 4$ of $g$ in ${\bold
Q}(\omega)$.\smallskip  \item{(e)}
 Suppose
that $K(\beta):K$ and $L(\gamma):L$ are simple algebraic extensions.
Give sufficient conditions on the minimum polynomials of $\alpha$ and $\beta$
under which an isomorphism of fields $\theta:K\rightarrow L$ can be extended to
an isomorphism of the two extensions.  Use your condition  to show that
for any integer $j$
with $1\leq j\leq 4$ there is an automorphism of 
${\bold Q}(\omega):{\bold Q}$ mapping $\omega$
to $\omega ^j$.\smallskip 
 \bigskip
 
   
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