Spring 2017: Math 541 (L001) Modern Algebra

Instructor: Manuel Gonzalez Villa, Van Vleck Assistant Professor.
villa@math.wisc.edu
Lectures: MWF 08:50-09:40 am, in Van Vleck B221
Office Hours: WeTh 14:00-15:00, in Van Vleck 719

Background and Goals:

This is the first semester of an introduction to basic abstract algebra. It is recommended for math majors and it is essential for students preparing for graduate studies in mathematics and in some related fields.

Course Content:

Group Theory Definition and basic properties, subgroups. Examples: Cyclic groups, matrix groups, unit groups, Dihedral groups, symmetric groups, etc. Group homomorphisms, cosets, normal subgroups, factor groups fundamental theorem of homomorphisms. Direct product and semi-product of groups. Cayley's theorem, Lagrange theorem. Conjugacy classes, Sylow's theorem, group action (if time permits).
Ring Theory Definition and basic properties, subrings. Examples: integers, Gausian integers, Z/n, polynomial rings, matrix algebra, etc. Ideals, quotient rings, ring homomorphisms, fundamental theorem of homomorphisms. Ideals, principal ideals, integral domains, PID, maximal and prime ideals Irreducible polynomials in a polynomial ring, division algorithm. Unique factorization, UFD.
Field Theory Definition and basic properties, subfields. Examples: Q, R, C, finite fields. Field extensions, number fields, in particular quadratic fields and cyclotomic fields.

Required reading:

M. Artin, Algebra, Pearson (Prenctice Hall), 2011, ISBN: 9780321998033.

Additional Recommed reading:

R. Dummit and R. Foote, Abstract Algebra, WILEY, 2004, ISBN: 9780471433347.
P. Aluffi, Algebra (Chapter 0), AMS, 2009, ISBN: 9781470411688

Exams:

Midterm: Friday, March 17 (in-class).
Final: Wednesday, May 10, 7:45-9:45 AM

Homework:

Homework 1, due on Friday, February 17.
Exercises 2.1, 2.3, 2.4, 3.1, 3.2, 4.1, 4.2, 4.5, 4.6, 4.10, 5.1, 5.2, 5.3, 6.2 and 6.5 from Chapter 2 of M. Artin's Algebra.
Homework 2, due on Friday, March 3, 2017.
Homework 3, due on Monday, March 13, 2017.