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\centerline{Mathematics 400--Directed Readings in Algebra}
\centerline{Course Plan--Spring 2020}
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{\bf Professor:}    John Little\smallskip\noindent
{\bf Office:}       331 Swords Hall\smallskip\noindent
{\bf Office Phone:} 793-2274\smallskip\noindent
{\bf Email:} {\tt jlittle@holycross.edu}
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{\bf Course Description}
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We plan to cover the same material as (one common version of)
the Modern Algebra 2 course when there is sufficient demand
and staffing for that to be offered.  This will fill a gap in John Graf's
preparation for pure mathematics graduate programs and 
give him some additional background.

 We will meet twice a week for 75-minute
sessions (exact schedule to be determined).  Each meeting
will involve some presentations of problem solutions by John, 
some introduction of new material, by me and possibly by 
John on occasion.  

The course will concentrate
on the theory of {\it rings and fields}, an area of abstract algebra that
grew historically out of the desire to understand the basic 
algebraic properties of arithmetic operations such as factorization
of integers and its generalizations, and the solution of algebraic
equations.  Our main goal is a fascinating subject called ``Galois Theory''
(after the unfortunate $19^{th}$ century French mathematician \'Evariste
Galois).  This theory gives a complete answer to the question whether a 
polynomial equation 
$$a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 = 0$$
can be solved by means of an radical formula in the coefficients 
(such as the {\it quadratic formula\/} in the case $n = 2$).  Galois 
Theory associates to such an equation 
a certain group of permutations of the roots of the equation, whose
algebraic structure determines whether a ``radical'' formula for 
the roots exists.  

The topics we will be studying are:
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\item{1.} The basic language of ring theory, ideals and homomorphisms
\item{2.} Factorization in rings, unique and otherwise 
\item{3.} Fields, algebraic and transcendental extensions, applications
to the ancient Greek construction problems (duplication of the cube, 
squaring the circle, trisecting a general angle).   
\item{4.} Galois Theory of equations 
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We will spend roughly equal amounts of time on each unit.  
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{\bf Text}
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We will use a standard advanced undergraduate algebra text such as
Artin, Herstein, or possibly the even more advanced text by 
Dummit and Foote (used in many 1st year graduate courses).  
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{\bf Assignments and Grading}
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The grading will be based on
\item{1.}  Weekly problem sets (40\%)
\item{2.}  Problem presentations (10\%)
\item{3.}  Larger midterm problem set (20\%)  
\item{4.}  Larger comprehensive final problem set (30\%)
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