Math 594: Algebra II

Professor: David E Speyer

Winter 2013

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.

Rough Syllabus

Problem Sets

Topics covered

Group theory:

Representation Theory:

Galois Theory