MAT 150C: Modern algebra. Spring 2020
Instructor: José Simental Rodríguez.
Teaching assistant: Matthew Littman.
This is a continuation of MAT 150AB. This time, the focus will be on fields and Galois theory, which roughly speaking studies symmetries on roots of polynomials. Along the way, we will need to develop quite a bit of ring theory, focusing on rings of polynomials and factorization of elements. I will also try to give connections of these topics to other areas such as topology, combinatorics and algebraic geometry.
Textbook: Algebra, 2nd edition, by M. Artin (Pearson), ISBN 978-0132413770. We will cover, roughly, chapters 12, 13, 15 and 16.
Lecture notes are available on Canvas.
Homework:
Week 1. Review of ring theory from MAT 150B.
Week 2. Basics of algebraic geometry. Unique factorization domains and Principal ideal domains.
Week 3. Factorization of integer polynomials. Gauss lemma and Gauss primes.
Week 4. Algebraic integers. Multiplication and factorization of ideals.
Week 5. Prime ideals. More algebraic geometry.
Week 6. Field extensions, algebraic and transcendental elements.
Week 7. Ruler and compass constructions. Symmetric functions and the discriminant.
Week 8. Splitting fields, fixed fields, separable and Galois extensions.
Week 9. The main theorem of Galois theory.
Week 10. Cyclotomic extensions. Unsolvability of the quintic.