The main topics of this course are finite group theory and Galois theory.
The goal for the group theory portion of the class is to be able to classify
all groups of a particular order. (For arbitrary orders, the solution is
unknown, but we will mainly consider orders less than 64.) Here, classification
means producing a list of groups, say G1,...,Gk, such that every group
of that order is isomorphic to exactly one of the listed groups.
To be effective, we shall also require tools that allow us to analyze a
particular group of that order and determine which of G1,...,Gk is
isomorphic to it.
In order to find the groups G1,...,Gk, we will find it valuable to
compute the automorphisms of particular groups H. Here, an automorphism
of H is a group isomorphism f: H→ H. The automorphisms of H form
a group under composition.
You have seen classification theorems before, in Math 524: The primary
rational form gives a classification up to similarity
for the matrices whose characteristic
polynomial has a particular prime decomposition. Here, the characteristic
polynomial plays a role analogous to that of the order of a group. Recall
also that the same theorem gives a classification of all abelian groups
of a given order.
The goal for the Galois theory portion of the class is to study finite
extensions of fields. Here, if F is a field, a finite extension of
F is a field E containing F such that E is finitely generated as
a vector space over F. We are particularly interested in the case where
E is what's called a Galois extension of F. In this case, we shall
compute what's called the Galois group, Gal(E/F), of E
over F, which is the group (under composition) of field isomorphisms
f: E→ E that restrict to the identity on F. Using this, we can
find all subfields of E containing F.
The grading will be based on an in-class final exam and a number of
problem sets.
You are strongly encouraged to come often to office hours. The material is
quite challenging, and is best learned in discussion. We will have group
work sessions in ES 135. There is a wonderful synergy from working together.
Everyone profits, including me. You are also encouraged to discuss the
material with other students and faculty. There is no such thing as too
many insights, and different people think differently. There will be
multiple different proofs possible for the various results, and it is
very useful to see different ones.
