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MATH 150C (Modern Algebra) – Spring 2021, UC Davis

Course description

This course is the third part of a three-quarter introduction to Algebra. Algebra concerns the study of abstract structures such as groups, fields, and rings, that appear in many disguises in mathematics, physics, computer science, cryptography, … Many symmetries ||can be described by groups (for example rotation groups, translations, permutation groups) and it was the achievement of Galois to distill the most important axioms (=properties) of groups that turn out to be applicable in many different settings. $We will discuss rings and fields, in particular the important concept of factorization in rings, and at the end discuss Galois$ theory.
The class is primarily based on Chapters 12, 14-16 of Artin’s book.

1. Factorization
factorization of integers and polynomials; unique factorization domains; principal ideal domains and Euclidean domains; Gaussian integers; primes; ideal factorization

2. Modules
definition of modules; matrices, ||free modules and bases; diagonalization of integer matrices; generators and relations for modules; structure theorem for Abelian groups; application to linear operators

3. Fields
examples; algebraic and $transcendental elements; field extensions$; finite fields; function fields; algebraically closed fields

4. Galois Theory
fundamental theorem of Galois theory; cubic equations; primitive elements; cyclotomic extensions

Best proof so far in class (thanks to Gwen!):