Math 120: Modern Algebra

Spring 2012

Tuesdays and Thursdays 11:00 AM - 12:15 PM in 380-380F.

This class will cover groups, fields, rings, and ideals. More explicitly: Groups acting on sets, examples of finite groups, Sylow theorems, solvable and simple groups. Fields, rings, and ideals; polynomial rings over a field; PID and non-PID. Unique factorization domains.

Math 120 will be a fast-moving, high-workload class. Many students interested in this material will find Math 109 more appropriate. This is also a Writing in the Major class.

Professor: Peter McNamara 380-382H. office hours: (Starts week 3) Mon 1:30-2:30, Tue 10-11 or by appointment.

Course assistant: Julio Guitterez. office hours: (starts week 2) Wed 1-4pm, 380-380T or by appointment.

Text: The official recommended text is Dummit and Foote's Abstract Algebra. This is not required. You may find other sources useful, for example Artin's Algebra (just to name one). They should not be hard to find online.

Problem sets. (20%) There will be weekly homework assignments, posted here. No lates will be allowed (so the graders can just grade one problem set at a time, and hence have a better chance of getting them back promptly). But to give everyone a chance to get sick, or have busy periods, the lowest problem set will be dropped.

Problem sets will be due Thursdays in class, or may be emailed to by 11am sharp local time. If emailing, please put "Math 120 homework" in the subject.

Homework 1.
Homework 2.
Homework 3.
Homework 4.
No Homework due week of Thu 10 May due to midterm.
Week of 17 May. Write your WIM drafts.
Homework 5. Due 24 May.

Writing in the major assignment. (15%) Students must complete a writing assignment. This component of the course has its own website.

Midterm exam (20%) In Class Tue 8 May, 11am sharp. You may take in up to (1/16)m2 of paper with notes (which may be written on both sides). Syllabus: Chapters 1 through 5 of Dummit & Foote +6.3 and -[anything we didn't cover]. Exam solutions.

Links to practice papers: midterm, midterm, final, final, final, midterm. Some of these midterms were earlier in the quarter than ours, so will cover less material.

Final exam (45%) Fri 8 June, 3:30pm-6:30pm. In 380-380F. You may take in up to (1/8)m2 of paper with notes (which may be written on both sides). Syllabus:as below. This is basically Ch 1-8 + 9.1,9.2,9.5,13.1,13.2. without 7.5, any noncommutative rings, and the detailed structure of nilpotent & solvable groups in Ch 6. Exam solutions: Q 2,3,4 and Q 1,5,6.

Tue Apr 3: welcome, definition of group, examples of groups (dihedral, symmetric, cyclic, general linear); isomorphism of groups. Words you should know: binary operation, associative, commutative, group, abelian, identity, inverse, group operation, cyclic group, symmetric (or permutation) group, dihedral group, general linear group.
Thu Apr 5: subgroups, group actions, orbit-stabaliser theorem.
Tue Apr 10: Homomorphisms, kernels, normal subgroups, sign homomorphism, alternating group.
Thu Apr 12: Burnside's Lemma, Lagrange's Theorem, Cosets, Subgroup generated by a set.
Tue Apr 17: Cyclic groups, Quotient groups, First Isomorphism Theorem.
Thu Apr 19: Semidirect products, simple groups, simplicity of An.
Tue Apr 24: Classification of finite abelian groups.
Thu Apr 26: Jordan-Holder Theorem, group presentations.
Tue May 1: Free groups, Sylow theorems.
Thu May 3: Sylow theorems continued.
Tue May 8: Midterm.
Thu May 10: Odds and ends about groups. Automorphisms of a cyclic group, p-gropus have centres, nilpotent groups.
Tue May 15: Solvable groups. Start of Ring theory.
Thu May 17: Ideals, quotients, 1st Isomorphism theorem, polynomial rings.
Tue May 22: Ideals of a quotient ring, Euclidean domains, Principal Ideal domains, Unique factorisation domains.
Thu May 24: Field extensions.
Tue May 29: Finite fields.
Thu May 31: End of finite fields. Ruler and compass constructions.
Tue Jun 5: Chinese Remainder Theorem.