History (somewhat idealized) and Overview: Since ancient Babylon and Greece, mathematicians have developed methods to solve systems of equations.
In the final years of the eighteenth century, mathematicians began to suspect that their tools were fundamentally incapable of solving certain equations: Gauss (1796)
asserted that the coordinates of a regular heptagon could not be found by straightedge and compass, and Ruffini (1799) gave an
incomplete proof that the roots of a general quintic polynomial could not be computed by basic arithmetic operations.A few decades after these questions were formally asked, they were answered. Abel (1832) proved Ruffini's claim and Wantzel (1837) proved Gauss's. In 1846, Galois invented what is now known as Galois theory, which clarified Abel and Wantzel's results and permitted many new ones. From Galois's perspective, Galois theory was a tool to systematically analyze the process of extracting roots of equations. In modern mathematics, it is a crucial tool in number theory, algebraic geometry and commutative algebra, as well as still being relevant to its original purpose. The subject of this course is Galois theory and the topics which lead up to Galois theory. Specifically, we will discuss group theory, representation theory, and field theory, and then move to the beauties of Galois theory itself. (I mean "lead into" logically, not historically. None of these topics was recognized as a separate topic in Galois's time and, indeed, Galois theory was a central factor in prompting their study.) Linear algebra could also be considered a topic leading into Galois theory. As abstract linear algebra is a prerequisite for this course, we will not cover it as a separate subject, but we will certainly use linear algebra often and I expect many of you will improve your knowledge of it. |

**Text:** *Abstract Algebra*, Dummet and Foote, ISBN
978-0471433347

My goal is to use the textbook as a reference for proofs of
any material I skim past in class, and to provide any necessary background.
I will try to provide handouts covering any material I discuss which
is not in the textbook.
A number of students have pointed out to me that Karen Smith's notes
on representation theory are an excellent resource.

**Office Hours:** Monday 10-12 in 2844 East Hall and Thursday 3-6
in the Math Common Area (East Hall second floor lounge). I intend for the Thursday office hours in particular to be an opportunity for you to discuss the problems with
each
other as well as me.

**Professor:** David E Speyer, 2844 East Hall, speyer@umich.edu

**Course homepage:** http://www.math.lsa.umich.edu/~speyer/594

**Level:** Graduate students and advanced undergraduates.

**Prerequisites:** Prior exposure to the definitions of groups,
rings, modules and fields. Abstract linear algebra over an arbitrary
field. Principal ideal domains (PIDs) and unique factorization. 513
and 593 are enough background; please speak to me if you have
questions about your background.

You are welcome to work together with your classmates

In particular, I will be in the Math Common Area from 3-6 on Thursdays; I encourage you to gather there to discuss problems both with me and each other.

I do not intend for you to need to consult other sources, printed or online. If you do consult such, you should be looking for better/other expositions of the material, not solutions to specific problems.

You

Problem Set 2 (TeX), due January 25. Themes: Semidirect products, short exact sequences, groups acting on groups Solution Set (TeX)

Problem Set 3 (TeX), due February 1. Themes:

Problem Set 4 (TeX), due February 8. Themes: Characters, actions on vector spaces Solution Set (TeX)

Problem Set 5 (TeX), due February 15. Solution Set (TeX)

Problem Set 6 (TeX), due February 22. Solution Set (TeX)

Problem Set 7 (TeX), due March 15. Theme: Elementary lemmas about polynomials. Solution Set (TeX)

Problem Set 8 (TeX) due March 22. Theme: Practice with splitting fields, derivations. Solution Set (TeX)

Problem Set 9 (TeX) due March 29. Theme: Using the Galois correspondence, transcendence degree. Solution Set (TeX)

Problem Set 10 (TeX) due April 8 (Monday!). Theme: Using the Galois correspondence, cyclotomic fields. Solution Set (TeX)

Problem Set 11 (TeX) due April 12. Solution Set (TeX)