MAT 1100: Algebra I, Fall 2016

Syllabus

• Group Theory: Isomorphism theorems, group actions, Jordan-Hoelder theorem, Sylow theorems, direct and semidirect products, finitely generated abelian groups, simple groups, symmetric groups, linear groups, solvable groups, free groups, generators and relations.
• Ring Theory: Rings, ideals, Euclidean domains, principal ideal domains, unique factorization domains, field of fractions.
• Modules: Modules, tensor products, modules over a principal ideal domain, applications to linear algebra.

Useful books for reference (I will not really follow any of them)

• Lang, Algebra, 3rd ed.
• Dummit and Foote, Abstract Algebra, 2nd ed.

• Paul Selick's course notes (around 2011)
• Dror Bar Natan's course page (2014)
• Rotman, Robinson both in the GTM series (for groups; they cover much more!)

Grading scheme

• Homework (5 assignments): 25%
• Term test: 25%
• Final: 50%

Homework (tentative schedule)

• Assignment 1, due Tue Oct 4
• Assignment 2, due Wed Oct 12
• Assignment 3, due Wed Nov 9
• Assignment 4, due Tue Nov 22
• Assignment 5, due Wed Dec 7

Links

• George Bergman's companion to Lang's Algebra
• Handout by Dror Bar Natan on simplicity of A_n (n \ge 5)
• An example of a PID that is not Euclidean. (There is a much nicer proof that this ring is a PID, but it requires some algebraic number theory.)
• Proof of the elementary divisor theorem. (See Theorem 7.1 and its proof in these notes.)