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\centerline{Mathematics 352 -- Abstract Algebra 2}
\centerline{Information on Final Exam}
\centerline{April 24, 2008}
\vskip 10pt
\noindent
{\it General Information, Format and Groundrules}
\vskip 10pt
The final exam for this course will be given during the 
regular scheduled period for 11:00 MWF classes -- 8:30am - 11:30am
on Saturday, May 3.   
This will be an {\it closed book} exam.  
The exam will be written so that if you are well prepared and 
work steadily, you should take about 2 hours to complete it, 
but you will have the full 3-hour exam period to work on 
the exam, so this should eliminate most of the element of 
time pressure.

\vskip 10pt
\noindent
{\it Topics}
\vskip 10pt
The final exam will be a {\it comprehensive exam}.  It will
cover all of the material we have discussed since the beginning of
the semester, giving roughly equal attention to the sections of 
the class corresponding to the two midterms and also the 
definition of solvable groups and the question of 
which polynomial equations are solvable by radicals.  

Like the midterm exams, this one will
focus mainly on the key definitions and theorems we have studied.
There will also be a few problems or parts of problems
where you will need to apply some of those definitions
and theorems to new situations.

\vskip 10pt
\noindent
The topics to be covered are:
\vskip 10pt
\begin{enumerate}
\item  The structure theorem for finite abelian groups: Every finite 
abelian $G$ is isomorphic to a direct sum of cyclic groups of prime power order.
Know the classifications of these via the \emph{elementary divisors} and the 
\emph{invariant factors} and how to go back and forth between both forms.
\item  The Sylow theorems and the background needed for their proof -- 
the conjugacy relation on a group, the class equation of $G$ and the role of the 
center $Z(G)$, centralizers of $a \in G$, the formula 
$|C_a| = [G:C(a)] = |G|/|C(a)|$ (that is: the order of the conjugacy class
of $a$ is equal to the index of the centralizer of $a$), 
normalizers of subgroups, and so forth.
\item  Applications of the Sylow theorems to structure of finite groups,
dihedral groups, etc.  
\item  Vector spaces over a field, including examples where the vector
space is an extension field.
\item  Extension fields, simple extensions (that is $F(u)$ for $u$), 
algebraic and transcendental elements over $F$, minimal polynomials of algebraic
elements, how to find $[F(u):F]$ when $u$ is algebraic over $F$.   
\item  Algebraic extensions and their properties.
\item  Splitting fields of $f(x) \in F[x]$ and normal extensions.
\item Separable polynomials and extensions, the Primitive Element
Theorem (Theorem 10.18).
\item  Finite fields, the prime field, constructing fields of 
order $p^n$ for all $n \ge 1$.  All fields of order $p^n$ are 
isomorphic.  A field of order $p^n$ has subfields of order $p^k$
if and only if $k\,|\,n$.  
\item  Geometric constructions, constructible numbers, the 
characterization of constructible numbers:  $\alpha$ is 
constructible if and only if $[{\bf Q}(\alpha) : {\bf Q}] = 2^m$ for
some $m$.  The three Greek construction problems and their resolution.
\item  $F$-automorphisms of an extension field $K$ of $F$,
the Galois group ${\rm Gal}_F K$,
\item  The Galois correspondence between subgroups 
of ${\rm Gal}_F K$ and intermediate fields $E$ with 
$F \subseteq E \subseteq K$, Galois ($=$ finite-dimensional, separable, and normal) 
extensions, the Fundamental Theorem of Galois Theory.
\item  Root towers, solvable groups, the Galois Criterion
for solvability by radicals.  Non-solvability of $S_5$ and $A_5$, and 
applications to quintic polynomials.   \emph{Note:}  Problems 11 - 18 in 
Section 11.3 are good here!
\end{enumerate}
\vskip 10pt
\noindent
{\it Review Session}
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I would be happy to schedule a day or evening review session before the exam, but
I have an off-campus commitment in the evening on Thursday, May 1, so that evening
will not work.
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\noindent
{\it Key Theorems and Proofs to Know}
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\begin{enumerate}
\item  Know the statement of the structure theorem for finite 
abelian groups (Theorem 8.7 in Hungerford).
\item  Know the statements of the three Sylow theorems and 
the proof of Sylow 1: If $p$ is a prime and $p^n | |G|$, 
then $G$ contains a subgroup of order $p^n$.
\item  Know the proof of the multiplicativity of degrees in
towers of field extensions:  If $F \subseteq E \subseteq K$ are
field extensions, and $[E:F]$ and $[K:E]$ are finite, then 
$[K:F] = [K:E][E:F]$ (Theorem 10.4 in Hungerford). 
\item  Know the proof of the main theorem on simple 
extensions (Theorem 10.7 in Hungerford):  $F(u) \simeq F[x]/(p(x))$
where $p(x)$ is the minimal polynomial of $u$ over $F$, and 
$[F(u):F] = n$ if $\deg p(x) = n$.  
\item  Know the proof that $[K:F]$ finite implies every
element of $K$ is algebraic over $F$ (Theorem 10.9 in Hungerford).
\item  Know how to show that the splitting field of 
$x^{p^n} - x$ over $F = {\bf Z}_p$ is a field of order $p^n$. 
(This is one implication from Theorem 10.25.) 
\item  Know the statement of the Fundamental Theorem of 
Galois Theory (Theorem 11.11 in Hungerford).   
Also know the statement and proof of the Lemma 
used in the proof of the Fundamental Theorem (Lemma 11.7 in Hungerford). 
\item  Know the statements and the proofs of the two theorems 
from class describing the structure of Galois groups of radical 
extensions.  Let $p$ be prime in both of the following:
\begin{enumerate}
\item  Let $F$ be a field containing a primitive $p$th
root of $1$, $\zeta_p$.  If $K = F(u)$ where $u^p = a$ for some $a \in F$
that is not a $p$th power in $F$, then ${\rm Gal}_F K$ is a cyclic group
of order $p$ (isomorphic to the additive group $(\Z_p,+)$).
\item If $\zeta_p$ is a primitive $p$th root of $1$, then 
${\rm Gal}_\Q \Q(\zeta_p)$ is a cyclic group of order $p-1$ (isomorphic
to the multiplicative group $(\Z_p^*,\cdot)$).  
\end{enumerate}
\end{enumerate}
\vskip 10pt
\noindent
{\it Philosophical Comments and Suggestions on How to Prepare}
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\begin{itemize}
\item  
The reason we give final exams  
in almost all mathematics classes is to give students
a final opportunity to ``put whole courses together'' in their minds. 
This course is an especially central one connecting many key 
ideas in mathematics and illustrating their historical development. 
And in fact, the ideas we have learned lead directly to many
areas of current research and recent ``big theorems'' in the 
subject.  
\item 
There is a lot here, but there are also many, many
connections between different 
portions of what we have done -- identifying them and 
understanding how they fit together is a good thing to do 
at some point.  (One small example:  How does the 
Structure Theorem for finite abelian groups show 
that for any finite abelian group $G$, 
there exists a chain of subgroups
\[
G = G_0 \supset G_1 \supset G_2 \supset \cdots \supset \{e\}
\]
where $G_i/G_{i+1}$ is cyclic of prime order for each $i$, hence
that $G$ satisfies one form of the definition of solvability?) 
\item 
Reread your class notes in addition to the
text, especially for topics where you lost points on the midterms.  
There are a lot of worked-out examples and discussions of all of 
the topics we have covered there.
\item 
Get started reviewing early
and do some work on this {\it every day} between now 
and the date of the final.  Don't try to ``cram'' everything in
at the end.  
\item  
Look over the midterm exams with the solutions. Go over your 
corrected problem sets.  If there were questions where you lost
a lot of points, be sure you understand why what you did was not
correct, and how to solve those questions. 
\item  Be sure you actually do 
enough practice problems so that you have the 
facility to solve exam-type questions in a limited
amount of time.   {\it Even if you 
have saved solutions for practice problems from the midterms},
it is going to be much more beneficial to do practice problems
starting ``from scratch'' rather than just reading old
solutions.  
\end{itemize}
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