MATH 5020: Topics in Algebra
Description: Advanced topics chosen from group theory, ring theory, number theory, Lie theory, combinatorics, commutative algebra, algebraic geometry, homological algebra, and representation theory. With change of content, this course may be repeated to a maximum of twelve credits.
Prerequisites: MATH 5211.
MATH 5020 - Section 1: Infinite Galois Theory
Description: This is a second course in Galois theory. We will begin with a review of the fundamental theorem of Galois theory that is taught in Abstract Algebra II and discuss a number of explicit examples in zero and positive characteristic (to explore separability issues), including finite fields. The course will emphasize both theoretical and computational methods (with software such as Sage and Magma) to study field extensions. Correspondences similar to the Galois correspondence show up in other areas of mathematics, such as in complex analysis and topology, and we will emphasize these similarities. The main goal is to discuss Galois theory of infinite extensions (esp. of finite fields, the rational numbers, and function fields). The usual Galois correspondence breaks down for infinite extensions, but can be saved by introducing topological ideas: infinite Galois groups are best viewed not as abstract groups, but as compact topological groups. Once again, we will work with explicit examples such as the (separable) closure of a finite field, or cyclotomic extensions. Time permitting, we will introduce Galois representations, and produce a few examples.
Prerequisites: MATH 5211 or consent of instructor
Sections: Fall 2017 on Storrs Campus
|14199||5020||001||Lecture||TuTh 02:00:00 PM-03:15:00 PM||MONT420||Lozano-Robledo, Alvaro|