# MATH 150B (Modern Algebra) - Winter 2018, UC Davis

## Course description

This course is the second 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 important classes of groups such as the linear groups and their group representations and introduce the notion of rings.
The class is primarily based on Chapters 8-11 of Artin's book.

1. Bilinear Forms
symmetric forms; orthogonality; the geometry associated to a symmetric form; hermitian forms; Spectral Theorem.

2. Linear Groups
the classical linear groups; the special unitary group; orthogonal representation of SU_2; SL_2

3. Group Representations
irreducible and unitary representations; characters; Schur's lemma

4. Rings and Fields
definition of rings and fields; formal construction of integers and polynomials; homomorphisms and ideals; quotient rings and relations in a ring; integral domains and fraction fields; maximal ideals