Instructor: Matt Kerr Office: Cupples I, Room 114 e-mail: matkerr [at] wustl.edu Office Hours: 12-1 M and 1-2 F Course Outline: I. Sets A few preliminaries.II. Groups Topics include: subgroups, Cayley and Lagrange theorems, group
actions and
Burnside's lemma, orbits and conjugacy classes, cosets, normal subgroups,
quotient groups, homomorphisms, Sylow theorems.III. Rings Topics include: integral domains, ideals, homomorphisms,
fields, polynomial rings, Euclidean algorithm, multiplicative group of
a finite field, principal ideal domains, unique factorization domains,
Gauss's lemma, irreducibility tests, algebraic number rings.IV. Modules Topics include: Schur's Lemma, structure theorems for finitely generated
modules over a PID and abelian groups, canonical forms, endomorphisms.V. Algebras Topics include: exterior algebras, division algebras, quaternions,
Frobenius and Wedderburn theorems.This is the first half of a year-long course which forms the basis for the Ph.D. qualifying examination in algebra. This semester we will cover primarily groups, rings, and modules. In the Spring semester we turn to Galois theory, then go more deeply into the structure theory of rings and algebras, and apply this to representation theory and commutative algebra. Prerequisites: Math 430 or the equivalent, or permission of the instructor. Class Schedule: Lectures are on MWF, from 10:00AM-10:50AM, in Lopata Hall Room 202. First class is Monday Aug. 26 and last class is Friday Dec. 6, with no class on Monday Sep. 2 (Labor Day), Monday Oct. 7 (Fall Break), or Wed/Fri Nov. 27/29 (Thanksgiving Break). Midterm Exam: TBA (covering groups and rings)Final Exam: Dec. 17, 10:30AM-12:30PM, in the usual classroom.Assignments: These will be due by Gradescope submission Tuesday by 5PM. Solutions will also be posted on Canvas and may include students' work. I encourage you to visit my office hours to discuss problem sets, and to form study groups to discuss the more difficult problems (though solutions must be written up independently). This is, of course, a "proof course". Please write your solutions legibly, linearly, logically and concisely, in complete sentences with proper mathematical grammar, and cite the results you use ("by Proposition (1.2.3) in the lecture notes/Jacobson, it follows that ..."). In addition, make sure to document any ideas that come from another text, person, or online source. Problem Set 1 (due Wednesday Sep. 4) Problem Set 2 (due Tuesday Sep. 10) Problem Set 3 (due Tuesday Sep. 17) Problem Set 4 (due Tuesday Sep. 24) Problem Set 5 (due Tuesday Oct. 1) Problem Set 6 (due Thursday Oct. 10) Grader: Zijing Zhuang, z.zijing [at] wustl.eduLecture Notes: These will serve as our primary text, supplemented by the Jacobson book (see below). I. Sets A. RelationsII. Groups A. IntroductionIII. Rings A. Examples of ringsIV. Modules A. Definition and examplesV. Remarks on Associative Algebras A. Algebras over a field Books: Nathan Jacobson, Basic Algebra I (2nd Ed.); Dover is the recommended textbook, which means that you should read some of it alongside
the notes (which loosely follow it) and some of the problems I assign will come from it.
Buy a copy of this (and possibly Basic Algebra II for next semester) -- it's cheap.
If you would like to read more adventurously than Jacobson and/or my lecture notes, here are some suggestions. First, there are many other excellent standard texts, including D.J.H. Garling, A Course in Galois Theory; Cambridge Univ. Press, Thomas Hungerford, Algebra; Springer-Verlag, Serge Lang, Algebra; Springer-Verlag, and Joseph Rotman, Advanced Modern Algebra; Prentice-Hall. Grading Policy:Your final grade for the semester is determined as follows: HW 40%, midterm 20%, final exam 40%. I will drop the lowest two grades you receive on homework. Homework and examination grades will be regularly updated on Canvas. Grades are typically curved in a course like this but will never be less than the following scale:
If you are a graduate student, a letter grade of B is required to pass; if you are an undergraduate taking this class Pass/Fail, you must earn a C- to pass. The Washington University academic integrity policies are here. All work submitted under your name is expected to be your own; please make sure to document any ideas that come from another source. Additional Resources:The linked document contains a wealth of information on university policy and resources for students. |