The focus of this course is the theory of groups. This is one of the core topics in pure mathematics, displaying beauty, elegance, simplicity and depth. The notion of a group developed over two hundred years ago, as a way to understand the symmetry of other mathematical objects. In the 20th century, group theory became one of the foundational topics in pure mathematics. By the end of the century, one of the major challenges in group theory was eventually solved, due to the work of dozens of mathematicians in hundreds of papers. By the end of this course you will understand what this challenge means: classify all finite simple groups. You will also appreciate some pieces of the answer.
In the last third of the semester we will study fields. There are two principal topics, constructing a field from an integral domain (Section 9.4) and constructing fields by adjoining roots of an equation (Chapter 10). The culmination of this portion of the course will be finite fields, which we will characterize completely. Finite fields are of great interest nowadays since they are used in error correction coding and cryptography.
The prerequisite for this course is Abstract Algebra 521A. You should already have some experience with fields from 521A, but we will review Chapters 4 and 5 as needed.
You should be conversant with the fundamentals of logic ("and", "or", "implies", "for all", "there exists", and negation). We will use a variety of proof methods in this course: direct proof, proof by contradiction, and mathematical induction. We use equivalence relations. This material is all covered in Discrete Mathematics, Math 245.
When we cover fields we will use some linear algebra, but just basic notions: linear independence, span, basis and dimension. We will spend some time in Section 10.1, which reviews this material.
I expect to spend about 9 weeks on groups, Chapters 7 and 8. We will continue with 6 weeks on fields, Section 9.4, and Chapter 10.
Written assignments should be carefully and neatly presented.
I expect to give three tests, but could be persuaded to substitute some sort of project for one of the test.
The relative weights of the work are given below.