**Honors Algebra: Math 512 and Math 513 **

Professor
Karen E. Smith

East Hall 3074

3-5048

Math 513 is the second semester of the year long sequenceMath 512/513 intended to introduce honors math students to the art and practice of abstract modern algebra.

Prerequisites:
Math 512.

The course meets MWF at 2 pm in 4088 East Hall.

Office hours immediately after class on mondays and 12 noon on Thursdays; also by appointment.

** Text: ** Artin, Algebra, first edition, 1991.

Stewart, Ian, Galois Theory, third edition. (actually, if you can
find it for a decent price, I like the second edition better).

** More recommended Reading: **

** 1.
** Dummit and Foote, Abstract Algebra (a book I seriously considered for Math 512/513; tell me what you think!)

2. Algebra with Galois theory, Author: Emil Artin ; notes by Albert A.
Blank., Publisher: Courant Institute of Mathematical Sciences, New York
University 2007

Syllabus for Math 513: We'll study rings and their modules. Modules are
common generalization of abelian groups and vector spaces. Two
applications of the theory will be the classification of finite abelian
groups and the Jordan canonical form for a linear transformation. We
will then study fields and Galois theory. Here some striking
applications are things you may have always wondered how to prove: there
is no ``formula in radicals" for the zeros of a polynomial of degree
five or more analogous to the quadratic formula, and you can not trisect
an angle with a compass and straightedge.

The course will be organized in the same way as Math 512

The Daily Update, a summary of what was discussed in class each day, including assignments and quiz announcements.

DJ's Notes from Lecture, starting after winter break. Please let DJ know if you have any feedback.

Quizzes: Quiz 1 , Quiz 2 , Quiz 3 , Quiz 4 , Quiz 5 , Quiz 6 ,

Problem Sets: Set 1 , Set 2 , Set 3 , Set 4 , Problems for Exam 1 , Set 5 , Set 6 , Set 7 , Set 8 , Set 9 , Friday, March 23, we will have an inclass exam when you turn this in. Set 10 , Set 11 , Set 12 , Information and Sample Problems for Oral Part of Final Exam

**513 Class Exercises: **

on quotient rings of polynomial rings;

on the Euclidean Algorithm;

Free modules and their homomorphisms;

Jordan Form and the Minimal Polynomial;

The proof of Jordan Form and Rational Canonical Form;

Practice Problems for Quiz 6;

Worksheet on Galois Theory;

Worksheet on Cyclotomic Extensions;

**512 Class Exercises: **

on the symmetry group of a square;

the quaternion group

the isometry group of Euclidean space

The Automorphism group of the quaternion group (assigned as part of Homework Set 5)

the dihedral groups

Orbits and Counting

Cayley Graphs

Representations of S4