MATH 150B (Modern Algebra) - Winter 2014, UC Davis

Lectures: MWF 3:10-4:00pm, 176 Chemistry (Anne Schilling)
Discussion Sessions: MAT 150B-001, CRN 69917, R 6:10-7:00 PM in HUTCHINSON 115
Instructor: Anne Schilling, MSB 3222, phone: 554-2326,
Office hours: Mondays 2:10-3:00pm, Wednesdays 11:10am-12:00pm
T.A.: Patrick Tam, MSB 3219
Office hours: Tuesdays, Thursdays 10:30-11:30am
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 Sage Online Notebook. 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 14 in class. The Final exam will be Wednesday, March 19, from 1-3pm.
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 SmartSite.
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

Problem sets

Homework 0: (voluntary) send me an email about yourself, your goal and expectations for the class or anything else you would like to share!

Homework 1: due January 17 in class: pdf

Homework 2: due January 24 in class: pdf

Homework 3: due January 31 in class: pdf

Homework 4: due February 7 in class: pdf

Homework 5: due February 14 in class: pdf

Midterm: February 14 in class: The midterm will cover all material covered in class from Artin Chapter 8 (Bilinear Forms), Artin Chapter 9 (Linear Groups), and Artin Chapter 10.1 (Definition of group representations), Chapter 10.2 (irreducible representations), Chapter 10.3 (unitary representations). In addition, one problem that was discussed in the Discussion session from the following list might be on the midterm (solutions are posted on SmartSite): Homework 6: due February 28 in class: pdf

Sage Example on characters: pdf

Homework 7: due March 7 in class: pdf

Homework 8: due March 14 in class: pdf

Final: Wednesday, March 19, 1-3pm in the usual classroom: The final will cover all material covered in class from Artin Chapter 8 (Bilinear Forms), Artin Chapter 9 (Linear Groups), Artin Chapter 10.1-10.8 (Group representations) and Artin Chapter 11.1-11.5, 11.8 (Rings). In addition, one problem that was discussed in the Discussion session from the following list might be on the final (solutions are posted on SmartSite):