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\centerline{Mathematics 352 -- Abstract Algebra II}
\centerline{Midterm Exam 2 -- April 16, 2008}
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\noindent
{\it Directions:}  Place all work you want to have considered for credit
in the blue exam book.  You have until 9:00pm to work on the exam if
you need that much time.  There are 100 regular and 10 Extra Credit points.  Some partial 
credit will be given for relevant definitions, statements of theorems, etc.  
Even if you cannot see how to complete some part, don't leave it blank.
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\noindent
I.  {\it Terminology.}
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\begin{enumerate}
\item[(A)] (5) What does it mean to say that a polynomial $f(x) \in F[x]$
is \emph{separable}?
\item[(B)] (5) What is an \emph{$F$-automorphism} of a field $K$ containing
a subfield $F$?
\item[(C)] (5) What does it mean to say that an extension field
$K$ of a field $F$ is \emph{Galois} over $F$?
\end{enumerate}
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\noindent
II. 
\begin{enumerate}
\item[(A)] (5)  State a necessary condition for the real number $r$ 
to be {\it constructible} by straightedge and compass,
starting from two given points $(0,0)$ and $(1,0)$.   
\item[B)] (10)  Given a \emph{sphere} of radius 1, is it possible
to construct the radius of a sphere with 5 times the volume of the original
sphere?  Why or why not?
\end{enumerate}
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\noindent
III.  Let $K$ be a finite field of order $81$.  
\begin{enumerate}
\item[(A)] (5) What is the characteristic of $K$?
\item[(B)] (10) Show that the mapping $\varphi : K \to K$ given by 
$\varphi(x) = x^3$ is an automorphism of $K$.
\item[(C)] (5) Does $K$ have a subfield of order $27$?  Why or 
why not? 
\end{enumerate}
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\noindent  
IV.  
\begin{enumerate}
\item[(A)] (15) State the Fundamental Theorem of Galois Theory. 
\item[(B)] (10) Show that if $K$ is a finite dimensional extension of $F$,
$H$ is a subgroup of ${\rm Gal}_F K$, and $E$ is the fixed field $E = E_H$, 
then $K$ is a simple, normal, separable extension of $E$.   
\end{enumerate}
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\noindent
V.  Let $K$ be a Galois extension of $F$ with ${\rm Gal}_F K = \langle \sigma\rangle$ 
a cyclic group of order 12.
\begin{enumerate}
\item[(A)] (10) What would the Galois correspondence look like in this case? 
(Construct the parallel diagrams of subgroups and intermediate fields, showing
inclusions, giving the intermediate fields other than $F$ and $K$ ``generic'' names 
like $E_1,E_2,$ etc.)
\item[(B)] (5) What are the degrees of each of the intermediate fields over $F$? 
\item[(C)] (10)  Let $E = E_H$ for $H = \langle \sigma^3\rangle$.  Is $E$ a 
normal extension of $F$?  Why or why not?  What is ${\rm Gal}_F E$?
\end{enumerate}
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\noindent
{\it Extra Credit}  (10) Let $K$ be the splitting field of the polynomial
$f(x) = x^3 - 2 \in \Q[x]$.  By Discussion 2, we know that $G = {\rm Gal}_\Q K \simeq S_3$.
What is the fixed field $E_H$ if $H$ is the cyclic subgroup of $G$ of order $3$?  (In $S_3$,
this is the subgroup generated by the permutation $(123)$.)  
Compute the fixed field using a basis for $K$ over $\Q$.
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