Math 611: Algebra I
Fall 2019

Instructor: Paul Hacking, LGRT 1235H, hacking@math.umass.edu

Meetings:

Classes: Mondays, Wednesdays, and Fridays, 12:20PM--1:10PM in LGRT 202.

Office hours: Mondays 3:00PM--4:00PM and Tuesdays 1:00PM--2:00PM in LGRT 1235H.

Course text: Abstract Algebra, by D. Dummit and R. Foote, 3rd ed., Wiley 2004. googlebooks.

Other useful references:

Introduction to commutative algebra by M. Atiyah and I. MacDonald. googlebooks.

The Todd--Coxeter algorithm (original paper) pdf.

Prerequisites: Undergraduate abstract algebra at the level of UMass Math 411--412.

Homework:

Homeworks will be assigned every 1--2 weeks and posted on this page.

HW1. Due Friday 9/20/19. Solutions.

HW2. Due Wednesday 10/2/19. Solutions.

HW3. Due Wednesday 10/16/19. Solutions.

HW4. Due Wednesday 11/13/19. Solutions.

HW5. Due Wednesday 12/11/19. Solutions.

Exams:

There will be one midterm exam and one final exam.

The midterm exam will be held on Wednesday 10/23/19, 7:00PM--9:00PM, in LGRC A201. Please try the midterm review problems here. Solutions pdf.

The midterm exam is here. Solutions are here.

The final exam will be a take home exam distributed on Wednesday 12/11/19 and due on Wednesday 12/18/19. The final exam is here.

The algebra sequence 611--612 is also assessed via the algebra qualifying exam. Syllabus.

Your course grade will be computed as follows: Homework 30%, Midterm 30%, Final 40%.

Overview of course:

Here is the syllabus for 611. Roughly speaking it correponds to Chapters 1--12 of Dummit and Foote. The full syllabus for the Graduate algebra sequence 611--612 is here.

(1) Group Theory.

Group actions. Counting with groups. p-groups and Sylow theorems. Composition series. Jordan-Holder theorem. Solvable groups. Automorphisms. Semi-direct products. Finitely generated Abelian groups.

(2) Rings.

Euclidean domain is a principal ideal domain (PID). PID is a unique factorization domain (UFD). Gauss Lemma. Eisenstein's Criterion.

(3) Modules.

Exact sequences. First and second isomorphism theorems for R-modules. Free R-modules. Hom. Structure Theorem for finitely generated modules over a PID. Rational canonical form. Jordan canonical form. Bilinear forms. Tensor product of vector spaces, Abelian groups, and R-modules. Right-exactness of the tensor product. Restriction and extension of scalars. Symmetric and exterior algebras.