This course is intended to complete your preparation for the preliminary examination in Algebra, which covers the material of this class together with that of Advanced Linear Algebra (Math 524). You are strongly encouraged to take the prelim as soon as possible after the completion of this course, when the material is still at your finger tips. I am very happy to discuss any questions you might have from past prelims during office hours.

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 $G_1,\dots ,G_k$, 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 $G_1,\dots ,G_k$ is isomorphic to it.

In order to find the groups $G_1,\dots ,G_k$, we will find it valuable to compute the automorphisms of particular groups $H$. Here, an automorphism of $H$ is a group isomorphism $f: H\to H$. The automorphisms of $H$ form a group under composition.

There are numerous such classification problems in the exercises in the book. You are
encouraged to tackle as many as possible. Some are also on old prelims.
I'm absolutely open to working with you on
any such problem.

You have seen classification theorems before, in Math 524: The rational canonical form gives a classification up to similarity for the matrices whose characteristic polynomial has a particular prime decomposition. I.e., two $n \times n$ matrices over a given field are similar if and only if they have the same rational canonical form. (Recall that the matrices $A$ and $B$ are similar if there exists and invetible matrix $P$ with $B=PAP^{-1}$.) Here, the characteristic polynomial plays a role analogous to that of the order of a group.
Recall also that the same underlying theorem (the Fundamental theorem of finitely generated modules over a PID.)
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 finite-dimensional 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 $f: E\to E$ that restrict to the identity on $F$. Using this, we can find all subfields of $E$ containing $F$.

Galois theory is especially important in number theory. It is also the basis for Galois' famous proof that there is no formula giving solutions to general polynomials of degree $n\ge 5$ in terms of rational functions of the coefficients.

The grading will be based as follows:

- We will have a collection of problem sets. Problem sets from previous semesters are posted on the course web and should give you a good idea of the level of difficulty of the problem sets you will face.
- There will be a sitdown final during finals week. It will consist of prelim level problems.

The final will count for 30% of your grade. The rest will come from the 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, either in my office or in ES 135, during office hours whenever you request them.
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.