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\centerline{Mathematics 352 -- Abstract Algebra II}
\centerline{Final Exam -- May 3, 2008}
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\noindent
{\it Directions:}  
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Place all work you want to have considered for credit
in the blue exam book.  You have until 11:30am to work on the exam if
you need that much time.  There are 200 regular and 20 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.}  Answer \emph{any five of the following}.  You can earn 
\emph{up to 10 points Extra Credit} for answering more than five.
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\begin{enumerate}
\item[(A)] (5) What is a \emph{Sylow $p$-subgroup} of a finite group $G$?
\item[(B)] (5) What is the \emph{class equation} of a finite group?
\item[(C)] (5) What does it mean to say that a point $Q$ in the plane 
is \emph{constructible}, starting from two given points $O = (0,0)$, 
and $P = (1,0)$?
\item[(D)] (5) What is the \emph{degree} of an extension field $K$
of a field $F$ (written $[K:F]$)?
\item[(E)] (5) What is the \emph{characteristic} of a field?
\item[(F)] (5) What is the \emph{Galois group}, ${\rm Gal}_F K$, of an
extension $K$ of a field $F$?
\item[(G)] (5) What does it mean to say that an extension field
$K$ of a field $F$ is \emph{Galois} over $F$?
\item[(H)] (5) What does it mean to say that a finite group $G$ is 
\emph{solvable}?
\end{enumerate}
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\noindent
II. 
\begin{enumerate}
\item[(A)] (10) State the Structure Theorem for finite abelian groups. 
\item[(B)] (20) Classify all finite abelian groups of order $n = 300$.  How many different
isomorphism classes are there?  Give both the elementary divisors and the invariant factors 
for each isomorphism class.
\end{enumerate}
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\noindent
III.
\begin{enumerate}
\item[(A)] (15) State the three Sylow Theorems for finite groups.
\item[(B)] (10) Show using the Sylow Theorems
that every group $G$ of order $n = 99$ has a non-trivial
normal subgroup $N$ (that is, a normal subgroup $N \ne \{e\}, G$). 
\item[(C)] (10 Extra Credit)  Show that every group of order $n = 99$
is isomorphic either to $\Z_9 \times \Z_{11}$ or to $\Z_3 \times \Z_3 \times \Z_{11}$.  
\end{enumerate}
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\noindent
IV. (10)  Show that if $K$ is an extension field of $F$ and $[K:F] < \infty$, 
then every element of $K$ is algebraic over $F$. 
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\noindent
V.  (15)  It can be shown, using some slightly tedious but elementary
manipulations with trig identities, 
that the number $x = 2\cos\left(\frac{2\pi}{7}\right)$ is a root of
the equation $x^3 + x^2 - 2x - 1 = 0$.  What does this imply about the constructibility
of the regular $7$-gon?  Explain.  
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\noindent
VI.  Let $K$ be the splitting field of $x^{64} - x \in \Z_2[x]$, an extension
field of $F = \Z_2$.
\begin{enumerate}
\item[(A)] (10) Show that the mapping $\varphi : K \to K$ given by 
$\varphi(x) = x^2$ is an $F$-automorphism of $K$.
\item[(B)] (10) Show that the mapping $\varphi$ from part (A) is an element
of order 6 in the Galois group ${\rm Gal}_F K$.  
(In fact, ${\rm Gal}_F K = \langle \varphi \rangle$ is cyclic of order 6.)
\item[(C)] (5) Find the orders of all the subfields of $K$.  
\end{enumerate}
VII.  
\begin{enumerate}
\item[(A)] (15) State the Fundamental Theorem of Galois Theory. 
\item[(B)] (20) Suppose we know that $K$ is a Galois extension of $F$ with
${\rm Gal}_F K$ an \emph{abelian} group of order $18$ whose Sylow $2$-
and $3$-subgroups are both cyclic.   How many intermediate fields $E$, 
$F \subset E \subset K$, are there and 
what are each of their degrees over $F$?
\end{enumerate}
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\noindent
VIII.  Let $p$ be a prime number and let 
\[
\zeta_p = \cos\left(\frac{2\pi}{p}\right) + i \sin\left(\frac{2\pi}{p}\right),
\]
be the primitive $p$th root of 1 in $\C$.  
\begin{enumerate}
\item[(A)] (10) Show that if $F$ is a field containing $\zeta_p$ and 
$K$ is the splitting field of the polynomial $x^p - a \in F[x]$ for some
$a\in F$ that is not the $p$th power of any element of $F$, then 
${\rm Gal}_F K$ is isomorphic to the additive group $(\Z_p,+)$ (cyclic of order $p$).  
\item[(B)] (10) Show that ${\rm Gal}_\Q \Q(\zeta_p)$ is isomorphic to 
the multiplicative group $(\Z_p^*,\cdot)$ (cyclic of order $p-1$).  
\end{enumerate}
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\noindent  
IX.  Let $\zeta_p$ be as in question VIII, and let $K$ be the splitting field
of $x^p - 2 \in \Q[x]$.  
\begin{enumerate}
\item[(A)] (10) Show carefully that $[K:\Q] = p(p-1)$.  
\item[(B)] (5)  Is ${\rm Gal}_\Q K$ a solvable group?  Why or why not?
\end{enumerate}
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\centerline{\it Have a safe, productive, and enjoyable summer!}
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