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{\bf \centerline{Math 362  Exam 2;  Fri April 4, 1997 12:55--1:45pm}}
\noindent{\bf Instructions.} Answer all  questions. You must show all
necessary working to receive full 
points for  a problem.  Calculators may be used if desired.
\bigskip
\noindent{\bf 1.} {\sl (15 points)}
\smallskip\item{(a)} Suppose that the point $(x,y)\in \real^2$ is
constructible (by ruler and compass) from the points $(0,0)$ and $(1,0)$. What
can one conclude about the field degrees $[\rat(x)\colon \rat]$ and
$[\rat(y)\colon \rat]$? \smallskip\item{(b)} Use the result stated in your
answer to (a) to give a careful proof that the point $(\root 5 \of  2, 0)$
 is not
constructible from $(0,0)$ and $(1,0)$.\bigskip

\noindent{\bf 2.} {\sl (35 points)}\smallskip\item{(a)}
Explain what is meant by saying that a finite field extension $L/K$  is (i)
the splitting field of a polynomial $f\in K[t]$ (ii) a normal extension 
(iii) a separable extension.
\smallskip\item{(b)} Under what conditions on the characteristic of the
field $F$ will any finite field extension $E/F$ be separable.?
Give an example of a  field extension which is finite and normal but not
separable.\smallskip\item{(c)} State the theorem giving the relationship between
finite normal extensions and splitting fields. Use this result to prove that if
$L/K$ is a finite normal extension and $M$ is an intermediate field, then $L/M$
is normal.\smallskip\item{(d)} Let   $L=\rat(\root 3 \of 5,e^{2\pi i/3})$,
$M=\rat(\root 3 \of 5)$ and $K=\rat$. Which of the extensions $L/K$,
$L/M$ and $M/K$ are (i) normal (ii) separable? \bigskip


  \noindent{\bf 3.} {\sl (25 points)} Let $L/ K$ be a finite
field extension. \smallskip \item{(a)} 
Give a
careful definition of the Galois group $G=\Gamma(L/ K)$.\smallskip
\item{(b)} For an intermediate field $M$, how is the corresponding subgroup 
$M^*$ of $G$ defined? Also define the  intermediate field
$H^{\dag}$ corresponding to a subgroup  $H$ of $G$.
\smallskip\item{(c)}  Prove that if $\sigma\in G$, then
$(\sigma H\sigma^{-1})^{\dag}=\sigma(H^{\dag})$, where
$\sigma H\sigma^{-1}$ is the subgroup of $G$ conjugate to $H$ by $\sigma$ and
$\sigma(H^{\dag})$ denotes the image of $H^{\dag}$ under $\sigma$.\bigskip  
  
\noindent{\bf 4.} {\sl (25 points)} You may assume in your answer to this
question that the polynomial $f=t^6+t^5+t^4+t^3+t^2+t+1\in \rat [t]$ is
irreducible. \smallskip\item{(a)} Let $\omega=e^{2\pi i/7}$. Explain why 
 $f$ is   the minimum polynomial of $\omega^i$ over $\rat$ 
for $i=1,\ldots,6$.  Conclude that $L:={\rat
}(\omega)$ is a splitting field of $f$ over $\rat$ and 
that for each $i=1,2,\ldots, 6$ there is a unique $\rat$-automorphism $\sigma_i$
of $L$ satisfying $\sigma_i(\omega)=\omega^i$.\smallskip
\item{(b)} Show that $\sigma_i\sigma_j=\sigma_{k}$ where
$0<k<7$ is the remainder on division of $ij$ by $7$. Conclude that the Galois
goup $\Gamma(L/\rat)$ is a cyclic group of order $6$ generated by $\sigma_3$.
\smallskip\item{(c)} Let $H$ be the subgroup of $G$ generated by  $\sigma_2$.
Find the degree of the extension $L/ H^{\dag}$.

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