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.

Course meets: Monday, Wednesday and Friday, 2:00-3:00 PM, 3096 East Hall

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,

Course homepage:

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.

Student work expected: I will give problem sets every week, due on Fridays, and take-home exams when we finish a topic.

Homework Policy: You are welcome to consult your class notes and textbook.

You are welcome to work together with your classmates provided (1) you list all people and sources who aided you, or whom you aided and (2) you write-up the solutions independently, in your own language. If you seek help from mathematicians/math students outside the course, you should be seeking general advice, not specific solutions, and must disclose this help. I am, of course, glad to provide help!

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. In particular, I will assign many standard lemmas which will be valuable for you to prove for yourselves; searching for other people's proofs of these results would be self-defeating.

You MAY NOT post homework problems to internet fora seeking solutions. Although I know of cases where such fora are valuable, and I participate in some, I feel that they have a major tendency to be too explicit in their help. You may post questions asking for clarifications and alternate perspectives on concepts and results we have covered.

Take home exams: On take-home exames, you may not consult with other people or outside sources; you may consult your notes and textbook, and the handouts I provide. There will be three take home exams, on Group Theory, on Representation Theory and on Galois theory. I currently anticipate that these will be due on February 1, March 1 and April 19; these dates may change.

Rough Syllabus

January: Introduction to group theory

Early February: Introduction to representation theory

Late February and Early March: Field extensions, solving equations before Galois

Late March and April: Galois Theory

Problem Sets

Problem Set 1 (TeX), due January 18. Themes: Group actions, normal subgroups Solution Set (TeX)
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: p-groups, commutators Solution Set (TeX) UPDATE: Solution to 5(e) TeX

Midterm 1 will be distributed on Friday, February 1 and due on Monday, February 4.

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)

Midterm 2 will be distributed on Friday, February 22 and due on Friday, March 1.

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)

Midterm 3 will be distirbuted on Friday, April 12 and due Friday, April 19.

Topics covered

All plans for future dates are subject to revision.

January 9: Solving quadratic, cubic and quartic equations (TeX)

Group theory:

January 11: Maps of groups, group actions. Reading suggestions: 1.6, 1.7, 3.1, 3.2
January 14: Normal subgroups Reading suggestions: 3.1, 3.3, Handout (for class discussion)
January 16: Short exact sequences, direct products, start of semidirect products. Reading suggestions 5.1, 5.4, 5.5
January 18: Semidirect products.
January 21: Martin Luther King Day, no class
January 23: The Jordan-Holder theorem (TeX)
January 25: Solvable groups, abelianization Reading suggestions: 6.1
January 28: Nilpotent groups Reading suggestions: 6.1
January 30: The Sylow theorems Reading suggestions: 4.5
February 1: Using the Sylow theorems Reading suggestions: 4.5, 6.2

Representation Theory:

February 4: Introduction to representations and characters Reading suggestions: 18.1
February 6: Examples of representations
February 8: Tensor products, wedge products
February 11: Uniqueness of direct sum decompositions Reading suggestions: 18.2
February 13: Irreducible and indecomposable modules; Maschke's theorem Reading suggestions: 18.1
February 15: Computing with characters 18.3
February 18: A group has finitely many representations; representations inside the regular rep.
February 20: Peter-Weyl for finite groups.
February 22: Applications of Peter-Weyl: Number of representations equals number of conjugacy classes; row and column orthogonality of the character table.
February 25: Using the ring of endomorphisms to decompose representations (TeX).
February 27: More on the ring of endomorphisms.
March 1: The representation theory of Sn (TeX).

Galois Theory

March 11 Review of important properties of polynomials. Reading suggestion: 9.1, 9.2, 9.3
March 13 Field extensions Reading suggestions: 13.1, 13.2
March 15 Straightedge and compass constructions Reading suggestions: 13.3, Construction of the 17-gon(TeX)
March 18 Splitting fields Reading suggestions: 13.4
March 20 Automorphisms of fields Reading suggestions: 14.1
March 22 Examples and applications
March 25 Algebraic closures Reading suggestions: 13.4 (Karen Smith lectures this day.)
March 27 More examples and applications
March 29 Solvable field extensions Reading suggestions 14.7
April 1: More on solvable field extensions
April 3: Algorithmically computing Galois groups Recommended reading: Chapter 13 of Cox, Galois Theory (link requires UMich login)
April 5: The fundamental Theorem of Galois theory (Karen Smith lectures) Reading suggestions: 14.2
April 8: The fundamental Theorem of Galois theory part 2
April 10: The normal basis theorem and the primitive element theorem (TeX)
April 12: Structure of a general algebraic extension Reading suggestions: 14.9
April 15: Inseparable extensions Reading suggestions: 14.9, Brian Conrad's notes
April 17: Transcendence degree
April 19: Algebraic independence and derivations
April 22: Something awesome