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\centerline{Mathematics 351 -- Abstract Algebra 1}
\centerline{Information on Exam 1}
\centerline{September 24, 2007}
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\noindent
{\it General Information, Format and Groundrules}
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The first midterm exam will be given
Monday evening October 1.  
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 beginning of
the semester, up to and including the material on the irreducible
polynomials in ${\bf R}[x]$ and ${\bf C}[x]$ from class
on Monday, September 24.  This is the material from 
Chapters 3 and 4 in Hungerford.

The exam 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{1)}  The definition of a ring and key examples.
\item{2)}  Basic properties of rings.
\item{3)}  Isomorphisms and homomorphisms.
\item{4)}  Polynomial rings, the division algorithm and consequences.
\item{5)}  Divisibility in $F[x]$.
\item{6)}  Irreducible polynomials and unique factorization.
\item{7)}  Roots of polynomials and factorizations.
\item{8)}  Results on irreducibility of polynomials in ${\bf Q}[x]$, 
${\bf R}[x]$, ${\bf C}[x]$.  
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\noindent
{\it Review Session}
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We will review for the exam in class on Monday, October 1.
Of course this means you will not have much time between then
and the exam to do further preparation.  {\it Begin your 
studying before the review session, please!}
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\noindent
{\it Key Theorems and Proofs to Know}
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\item{1)}  Know how to show that a given subset of a ring $R$ is, or is not, a subring
of $R$.  (Note: in most cases, the criterion of Theorem 3.6 in Hungerford is the
``preferred method.'')
\item{2)}  Know how to show that a given mapping $\varphi : R \to S$ is, or is not,
a homomorphism of rings.  Same for isomorphisms.
\item{3)}  Know how to prove the uniqueness of the quotient and the remainder
in polynomial division, given the existence.
\item{4)}  Know the statement and proof of the Unique Factorization Theorem in $F[x]$
(see Theorem 4.13 in Hungerford, and the class notes).
\item{5)}  Know how to prove that $a \in F$ is a root of $f \in F[x]$ if
and only if $(x - a)|f$.
\item{6)}  Know the statement and the proof of the Rational Root Test (Theorem 4.20 in Hungerford
and the class notes).
\item{7)}  Know the statement and proof of Eisenstein's Irreducibility Criterion in ${\bf Q}[x]$
(see Theorem 4.23 in Hungerford, and the slightly different proof in the class notes).
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\noindent
{\it Suggested Review Problems}
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From Hungerford:  
\item{} 3.1/5, 19, 38; 
\item{} 3.2/13, 14, 15, 21 (read page 59 first!), 26;
\item{} 3.3/13, 16, 18, 31; 
\item{} 4.1/5bd, 6, 19; 
\item{} 4.2/7, 8, 10; 
\item{} 4.3/7, 10, 12, 13, 25, 26;
\item{} 4.4/3, 14, 17;
\item{} 4.5/1bd, 10, 16, 17, 18.
\item{} 4.6/2, 3, 6.  
\bye

