**Modern Algebra II **

**Math 627B**

Course number: 19891

Spring 2005

Meeting MW 2:00-3:15

GMCS #308

San Diego State University

Final Exam: Mon. May 16, 1:00-3:00

**Professor**: Mike O'Sullivan

**Web page**: http://www.rohan.sdsu.edu/~mosulliv

**Email**: mosulliv@math.sdsu.edu

**Office**: GMCS #579, ext. 594-6697

**Office Hours**: M 3:15-5:00, W 3:15-5:00, F 1:00-2:30.

Other times: by appointment.

## Detailed Information

## Course Description

We will study algebraic geometry, one of the richest and oldest areas of
mathematics. During the 20th century, the theoretical and very
abstract side of the subject was prominent, but with the availability
of computers, the computational roots have been reinvigorated.
This course will develop the theory behind the computational tools.

What is algebraic geometry? Think back to high-school algebra where
you graphed polynomial equations and perhaps found the interestection
of plane curves defined by polynomials.
Now think about higher dimensional space and look at intersections of
hyper-surfaces defined by polynomial equations.
What is the dimension? How many components are there? What is the
simplest way to describe the intersection? These are some
fundamental questions of algebraic geometry.

The textbook we are using is excellent. I hope to cover the core
material, chapters 1-4, and then let student interest guide the rest
of the semester. There will be homework assignments with proofs and
compational exercises. I expect to incorporate computer assignments
as well. There will also be a final project, with a great deal of
latitude in choice of topic. You may focus on theoretical questions,
implementation of an algorithm, an applied problem, or some
combination. You may also develop an educational module for advanced
high-school students.

## Required Materials

Cox, Little, O'Shea * Ideals, Varieties, and Algorithms: An
Introduction to Computational Algebraic Geometry and Commutative
Algebra* 2nd Ed., Springer-Verlag, 1997.

## Prerequisites

A good understanding
of the basics of groups, rings and fields (Math 521A and 521B is enough).
In particular you might want to review the following.
- Polynomial ring in one variable: The division theorem, greatest
common divisor, Euclidean algorithm, the correspondence between
factors and roots, unique factorization.
- Integral Domains: ideals, the quotient of an integral domain by an
ideal, homomorphisms. The field of quotients of an ideal (I will cover this).
- Linear Algebra: nullspace, subspace, dimension, basis.