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\centerline{Mathematics 352 -- Abstract Algebra 2}
\centerline{Information on Exam 2}
\centerline{April 8, 2008}
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\noindent
{\it General Information, Format and Groundrules}
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The second midterm exam this semester will be given on
Wednesday evening, April 16.  
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 first exam this
semester, up to and including the material on the Fundamental
Theorem of Galois Theory 
from class on April 9 (Problem Set 9).  This is the material from
problem sets 6 through 9:  Sections 10.5, 10.6, Chapter 15,
11.1, and 11.2.  ({\it Important Note:} You will see, though,
that much of this material makes heavy use of concepts from 
the earlier sections of Chapter 10, and other things we 
studied earlier in the class, so there is a certain degree
of ``cumulativeness'' built into this material!) 
Like the other 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.  

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\noindent
The topics to be covered are:
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\item{0)}  The most important concepts from 
Sections 10.1-10.4 for this exam are:
Extension fields, simple extensions (that is $K = F(u)$ for some 
single $u$ in $K$), 
algebraic and transcendental elements over $F$, minimal polynomials 
of algebraic elements, how to find $[F(u):F]$ and a basis for $F(u)$ 
over $F$ when $u$ is algebraic over $F$, algebraic extensions and 
their properties.  Splitting fields of $f(x) \in F[x]$ and normal 
extensions.
\item{1)} Separable polynomials and extensions, the Primitive Element
Theorem (Theorem 10.18).
\item{2)}  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{3)}  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{4)}  $F$-automorphisms of an extension field $K$ of $F$,
the Galois group ${\rm Gal}_F K$
\item{5)}  The Galois correspondence between subgroups 
of ${\rm Gal}_F K$ and intermediate fields $E$ with 
$F \subseteq E \subseteq K$, Galois extensions, 
the Fundamental Theorem of Galois Theory.
\vfill\eject
\noindent
{\it Review Session}
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I would be happy to schedule an evening review session before 
the exam.  Monday, April 14 would probably be the best day for that.  
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\noindent
{\it Key Theorems and Proofs to Know}
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\item{1)}  Know the statement of the Primitive Element Theorem 
(Theorem 10.18).
\item{2)}  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{3)}  Know the statement of Theorem 15.6
in Hungerford (constructible numbers and quadratic towers), 
and know how to prove the characterization that if $\alpha$ is 
constructible, then $[{\bf Q}(\alpha) : {\bf Q}] = 2^m$ for
some integer $m\ge 0$ (see the class notes for this; Hungerford
does not state this particular fact).  The converse is also true here.
\item{4)}  Know the proof of Theorem 11.3 in Hungerford:
If $K$ is the splitting field of $f(x) \in F[x]$ and $u,v$
are elements of $K$, then there exists $\sigma \in {\rm Gal}_F K$
with $\sigma(u) = v$ if and only if $u,v$ have the same minimal
polynomial (know the part of the proof given in the previous 
Theorem 11.2, and how Theorem 10.14 applies here).  
\item{5)}  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).
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\noindent
{\it Suggested Practice/Review Problems}
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\item{$\bullet$} 10.5/9,11,12b
\item{$\bullet$} 10.6/9,10,16
\item{$\bullet$} Chapter 15/3,5,6,14 (and show that if $c > 0$ is constructible, then
$\sqrt{c}$ is also constructible)
\item{$\bullet$} 11.1/1,4,5,6,11,13,14,17
\item{$\bullet$} 11.2/5,6,14
\item{$\bullet$} Exhibit the Galois correspondence for 
\itemitem{(1)} $K = {\bf Q}({\root 3\of 3})$, $F = {\bf Q}$
\itemitem{(2)} $K = $ the splitting field of 
$f(x) = x^3 - 3 \in {\bf Q}[x]$, $F = {\bf Q}$.
\bye

