Math 593: Graduate Algebra I

Professor Karen E. Smith
East Hall 3074

FALL 2014
MWF at 2 pm in 1372 East Hall.
Office hours Tuesday and Thursday at 2-3; also by appointment.

Math 593 is the first course in the two-course required core in algebra for Math PhD students.

Prerequisites: Admission to the PhD program. Others should have completed Math 493-494, honors abstract algebra I and II.

Why I am asking you not to use laptops in class

Hoffman-Kunze Linear Algebra Text reference for bilinear forms (see chapter 10).

ALTERNATIVES: For PhD students who have already passed the algebra QR: Math 614 (commutative algebra), Math 631 (algebraic geometry) or Math 676 (algebraic number theory) are all beta-level algebra courses. I do not recommend topics courses for stage I PhD students. For masters students and PhD students who have not passed the algebra QR: Math 493-494 is a sophisticated course covering a lot of the same material of 593-594 without assuming students have already a year of advanced algebra. Math 575 (algebraic number theory) or 565 (graph theory) are other options. So is Math 420 (theoretical linea algebra). For undergraduates: Math 493-494 is a demanding honors algebra sequence. Math 420 (Abstract Linear Algebra) is a course I think all math majors, especially honors math majors should take (296-396 short-changes linear algebra). Math 412 (abstract algebra) is more appropriate for most math majors.

Text: Lang (Algebra), Dummit and Foote (Abstract Algebra), and Aluffi (Algebra: Chapter 0).

First Assignment DUE FIRST DAY OF CLASS: Read Lang Chapter I Section 11 (on categories and functors), and Chapter 2 Section 1 (on rings, this should be review). Also skim around the group chapter, as well as the first 7 chapters of Dummit and Foote: this should be mostly review. If not, take Math 493 instead.

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

Student Generated T/F quiz study materials for final exam: Gu , Vargo , Irvine , Zhu , Varma , Pagi, Springer , Lutz ,

Quizzes: Quiz 0 (solutions) , Quiz 1 (solutions) , Quiz 2 (solutions) , Quiz 3 , Quiz 4 , Quiz 5 , Quiz 6 ,

Exam 1 October 17, 2014 in class.
The final is as scheduled by the registrar on Dec 19.

Problem Sets: Set 1 ( Solutions ), Set 2 ( Solutions ), Set 3 ( Solutions ), Set 4 ( Solutions ), Set 5 ( Solutions ), Some remarks on Exam 1 , Set 6 ( Solutions ), Set 7 ( Solutions ), Set 8 ( Solutions ), Set 9 ( Solutions ), Set 10 ( Solutions ), Set 11 ( Solutions ), Set 12 and Take Home Quiz

Syllabus for Math 593: Graduate level algebra, including using universal properties to understand localization of rings and modules, tensor products, alternating products, etc. Structure theory for modules, especially over a PID, with applications to slick proofs of familiar results on the classification of finitely generated abelian groups, and Jordan and rational canonical form. Bilinear algebra. Some homological algebra. In terms of lectures, this is basically chapters 2, 3, 15, 16 of Lang, but I won't always follow Lang exactly and many of the other topics in Lang will be mentioned/reviewed in the exercises, etc. [Taken together, 593/594 will cover much of the material from Lang's book; Students may find Dummit and Foote or Aluffi more friendly reading, though possibly not quite sophisticated enough.]

Expectations: This will be a "seriously graded" course, moving at a fast pace with a high workload. Weekly problem sets are required, with just a few (random) problems carefully commented on (depending on the size of the class). Student backgrounds vary widely, and I will not teach to the lowest common denominator. For this reason, I can not expect all students to be able to do all problems: Examples will be drawn from many branches of mathematics, and the exercises will delve into to special topics that may be challenging for less experienced students. Weekly quizzes (25% of computed grade) will test whether students understand the most basic and core parts of the homework and lectures. If you have trouble telling which of the many exercises are ``core," please come ask advice. Midterm and final in class exams will account for 25% and 50% of the grade, respectively. Attendance is required, as is participation in class discussion.

Grades: The course is seriously graded on the Rackham scale, so students who actually show up and do the work will be graded on a scale of C to A+. Students who blow off the course will receive a D or F. There is no curve: in theory, everyone could get an A or everyone could get a C, but in practice, students usually distribute out so that a B+ or so is the median grade.

About homework: Ideally, these will be texed-up. Students may work together provided collaboration is acknowledged. Students may also turn in "joint" problem sets, but must follow the guidelines discussed in class. Use of websites such as MathStackExchange for "hints" on homework is considered CHEATING and will be promptly and vigorously discplined. In particular, a grade of E for the course for cheating will be assigned, even if the assignment itself was not worth a lot of points. Students who are aware of others cheating are *expected* to report this behavior.