Algebra Syllabus

Recommended Texts:

1. Lang (a fairly encyclopedic reference)
2. Atiyah and Macdonald (for the 3rd quarter)

FALL

Group Theory

This will include a rapid review of undergraduate material, as well as some material that is typically not taught at the undergrad level, such as semi-direct products, an introduction to defining groups by generators and relations, and the related ideas of free groups and amalgamated products. (Some part of the latter topics may also be covered in algebraic topology, and so the instructors for this course and that one might want to coordinate slightly what they teach.)

Galois Theory, including characteristic p and non-finite theory

As part of this, some basic undergraduate field theory will be rapidly reviewed, as well as some topics typically not covered at the undergrad level, such as a careful treatment of the notion of separability, and the basic theory of inseparable extensions. The most novel topic here is probably Galois theory for infinite extensions, and the connection with profinite group theory.

Profinite Groups, related group theory

This is probably best developed in tandem with the related Galois theory of infinite extensions. At the instructors discretion, related topics such as p-adic numbers might be also be included.

WINTER

• Rings and Modules
• Modules over PID's, including applications to linear algebra (via the theory of modules over polynomial rings), and other examples and applications (e.g. representation theory of cyclic groups)
• Artinian Rings and Wederburn theory
• Representations of finite groups, including character theory
• Tensor products (universal properties = bilinear maps)

SPRING

• Prime Ideals
• Spec
• Noetherian Rings
• Hilbert Basis Theorem
• Krull Dimension
• Local Rings
• Nakayama's Lemma
• Localizations
• Completions and the Artin-Rees lemma
• Minimal primes and radicals, nilradical
• Nullstellensatz