\magnification=\magstep1
\centerline{Mathematics 352 -- Abstract Algebra 2}
\centerline{Information on Exam 2}
\centerline{February 20, 2007}
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\noindent
{\it General Information, Format and Groundrules}
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The first midterm exam this semester will be given on
Wednesday evening, February 27.  
This will be an {\it closed book} exam.  It will 
start at 7:00pm, and you will have until 9:00pm
to work on it.  The exam will be written so that
if you are well prepared and work steadily, you 
should take about 1 hour to complete it, but you 
will have the extra time to eliminate any time pressure.

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\noindent
{\it Topics}
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The exam will
cover the material we have discussed since the start of
the semester, up to and including the material on splitting fields and normal extensions
from class on February 20 (Problem Set 5).  This is the material from
problem sets 1 through 5:  Sections 8.2, 8.3, 8.4, 8.5, 10.1, 10.2, 10.3, and 10.4. 
Like the exams last semester, 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.

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\noindent
The new topics to be covered are:
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\item{1)}  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 elementary divisors and the invariant
factors and how to go back and forth between both forms.
\item{2)}  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{3)}  Applications of the Sylow theorems to structure of finite groups,
dihedral groups, etc.  
\item{4)}  Vector spaces over a field, including examples where the vector
space is an extension field.
\item{5)}  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{6)}  Algebraic extensions and their properties.
\item{7)}  Splitting fields of $f(x) \in F[x]$ and normal extensions.
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\noindent
{\it Review Session}
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I would be happy to schedule an evening review session before the exam.
Monday, February 25 would probably be the best choice.  
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\noindent
{\it Key Theorems and Proofs to Know}
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\item{1)}  Know the statement of the structure theorem for finite 
abelian groups (Theorem 8.7 in Hungerford).
\item{2)}  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{3)}  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{4)}  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{5)}  Know the proof that $[K:F]$ finite implies every
element of $K$ is algebraic over $F$ (Theorem 10.9 in Hungerford).   
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\noindent
{\it Suggested Practice/Review Problems}
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\noindent
From Hungerford: 
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\item{} 8.2/3, 5, 6, 7, 14, 15, 16; 
\item{} 8.3/7, 12, 14, 15; 
\item{} 8.4/5, 8, 14, 15;
\item{} 8.5/2, 3, 14, 15;
\item{} 10.1/23, 24, 37;
\item{} 10.2/8, 9, 10, 11, 21, 22;
\item{} 10.3/4, 6, 9, 10, 13, 14;
\item{} 10.4/3, 4, 5, 9, 10, 17; 
\bye

