Math 594: Graduate Algebra II

Professor Karen E. Smith
East Hall 3074

Winter 2015
MWF at 2 pm in 4088 East Hall.
Office hours Tuesday and Thursday at 2-3; also by appointment.

Math 594 is the second course in the two-course required core in algebra for Math PhD students.

Prerequisites: Admission to the PhD program in pure mathematics, or Math 593. Others need permission from the instructor.

Representation Theory Notes

Text: Dummit and Foote (Abstract Algebra for some of the group theory), Emil Artin's "Modern Higher Algebra" for the field theory and Galois theory. (Ian Stewart's Galois theory text is also nice). I will also use my own lecture notes on representation theory. Michael Artin's book (Algebra) is also useful for much of the representation theory. Keith Conrad's online notes are also an excellent online source for Galois theory.

The Daily Update, a summary of what was discussed in class each day, including assignments and quiz announcements.

Worksheets: The Dihedral Group (Jan 9), The Quaternion Group (Jan 9), The Braid Group (Jan 9), Group Actions (Jan 11), Conjugacy classes in the Symmetric Group (Jan 14), The Class Equation (Jan), Characters (Feb 11) The Icosahedral Group (Feb 16), The Representations of S5 (Feb 18), Study T/F Questions for Exam 2.

Quizzes: Quiz 1 , Quiz 2 , Quiz 3 , Quiz 4 , Quiz 5 , Quiz 6 , Quiz 7 ,

EXAMS: Final , Exam 2 , Exam 1 (Solutions)

Study Questions and the Exam format.

The final will be an oral exam, scheduled during Exam period, at time convenient to us both.

Problem Sets: Set 1 ( Solutions ), Set 2 ( Solutions ), Set 3 ( Solutions ), Set 4 ( Solutions ), Set 5 ( Solutions ), Set 6 ( Solutions ), Set 7 ( Solutions ), Set 8 ( Solutions ), Set 9 ( Solutions ), Set 10 ( Solutions ), Set 11 ( Solutions ),

Syllabus for Math 594: Graduate level algebra, focusing on group actions on sets. We will review the action on a group on itself in various ways, leading to the class formula and Sylow's theorems [Dummit-Foote Chapters 4-6], then go through the basics on representation theory of finite groups (character tables, etc). We will cover the basic theory of representation of a finite group acting on a vector space. We will cover field theory and Galois theory, including the theory of field of prime characteristic.

Expectations: This will be a "seriously graded" course, moving at a fast pace with a high workload. Regular surprise quizzes will test whether students have absorbed the basic concepts and homework problems. The grading scheme is as follows: Quizzes (20%), first midterm (20%), second midterm (20%), final (40%). Attendance, homework and class participation required.

Grades: There is no curve, but unlike some honors or topics courses it is unlikely that everyone will get some form of A. In Math 594, the grade of B is perfectly respectable and denotes significant mastery. In practice, students without the prerequisites move to other courses and the remaining students work very hard, distrubuting out naturally so that a B+ or so is the median grade. Indeed, B+ was the median in 593.

Unlike Math 593, quiz and test rewrites will no longer be regraded. The reason is that some students reported in evaluations that this was percieved as unfair to less-prepared students, who felt then obliged to do more in the rewrite without being sure how to "maximize their points." My only goal with the rewrites was to give some concrete incentive to master the material, but if an illusion of some unfair system stacked against lower-scoring students was created, I apologize for that. In any case, the new policy is that now underperforming students (or anyone dissatified with their grade) should come discuss the material with me personally, to ensure mastery; no points will be given back.

About homework: Weekly problem sets are required. Ideally, these will be texed-up; if so (or if very carefully handwritten) they will be read to give you feedback. No numerical scores will be assigned to cut back on temptation to use websites such as stack exchange, which will be considered CHEATING. Students may work together provided collaboration is acknowledged. Students may also turn in "joint" problem sets.