MATH 150B (Modern Algebra) - Winter 2018, UC Davis

Lectures: MWF 3:10-4:00pm, Haring 2016 (Anne Schilling)
Discussion Sessions: MAT 150B-001, CRN 60662, R 6:10-7:00pm in Chemistry 176
MAT 150B-002, CRN 60663, R 7:10-8:00pm in Chemistry 176
Instructor: Anne Schilling, MSB 3222, phone: 554-2326,
Office hours: Mondays 4:10-5:00pm
T.A.: Gicheol Shin, MSB 3129
Office hours: Tuesdays 2-3pm, Thursdays 3-4pm
Text: I will mostly follow Michael Artin, Algebra, published by Pearson, second edition, 2011.
Another good reference is Dummit and Foote, Abstract Algebra, ISBN 0-471-36857-1.
Pre-requisite: MAT 150A
Problem Sets: There will be weekly homework assignments due on Fridays at the beginning of class.
You are encouraged to discuss the homework problems with other students. However, the homeworks that you hand in should reflect your own understanding of the material. You are NOT allowed to copy solutions from other students or other sources. If you need help with the problems, come to the discussion session and office hours! The best way to learn mathematics is by working with it yourself. No late homeworks will be accepted. A random selection of homework problems will be graded.
Computing: During class, I might illustrate some results using the open source computer algebra system Sage. When you follow the link, you can try it out yourself using CoCal. Or you can sign up for a Class Account with the math department. Log into and type the command `sage` to launch a Sage session in the terminal.
Exams: There will be one Midterm on Friday February 9 in class. The Final exam will be Thursday, March 22, from 10:30am-12:30pm.
There will not be any make-up exams!
Grading: The final grade will be based on: Problem sets 30%, Midterm 30%, Final 40%.
Grades will be recorded on Canvas.
Bed Time Reading: If you would like some bedtime reading related to math, I can recommend two really good books by Simon Singh: "Fermat's Last Theorem" and "The Code Book".

Course description

This course is the second part of a three-quarter introduction to Algebra. Algebra concerns the study of abstract structures such as groups, fields, and rings, that appear in many disguises in mathematics, physics, computer science, cryptography, ... Many symmetries can be described by groups (for example rotation groups, translations, permutation groups) and it was the achievement of Galois to distill the most important axioms (=properties) of groups that turn out to be applicable in many different settings. We will discuss important classes of groups such as the linear groups and their group representations and introduce the notion of rings.
The class is primarily based on Chapters 8-11 of Artin's book.

1. Bilinear Forms
symmetric forms; orthogonality; the geometry associated to a symmetric form; hermitian forms; Spectral Theorem.

2. Linear Groups
the classical linear groups; the special unitary group; orthogonal representation of SU_2; SL_2

3. Group Representations
irreducible and unitary representations; characters; Schur's lemma

4. Rings and Fields
definition of rings and fields; formal construction of integers and polynomials; homomorphisms and ideals; quotient rings and relations in a ring; integral domains and fraction fields; maximal ideals