# 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.

## 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.