# MA4H8 Ring Theory

**Lecturer:** Marco Schlichting

**Term(s):** Term 2

**Status for Mathematics students:** List C

**Commitment:** 30 lectures

**Assessment:** 3 hour exam (100%): 4 out of 5 problems, first is compulsory – 40%, the remaining problems are – 20% each

**Formal registration prerequisites**: None

**Assumed knowledge**: MA377 Rings and Modules

**Useful background: **MA3D5 Galois Theory and the p-adic numbers/Hasse principle part of MA257 Introduction to Number Theory will be useful

**Synergies: **The material of Ring Theory has many basic and more advanced links with Non-commutative Algebra and Algebraic Geometry, Algebraic Number Theory and Group Theory

**Content**:

- Review of MA377 Rings and Modules
- Jacobson Radical, Artin-Wedderburn, Hopkins theorem
- Central simple Algebras and the Brauer group
- Computation of the Brauer group of the rational numbers
- K_2 and the Brauer group of a field

**Aims**: The broad goal is to classify (non-commutative) Artinian rings, for instance, finite dimensional algebras over a field. We will show that every Artinian ring modulo a certain nilpotent ideal, called the Jacobson radical, is semi-simple. From MA377 Rings and Modules we know that semi-simple rings are finite products of matrix rings over division rings. Every division ring is an algebra over its centre which is a field. Under a suitable operation, the "set" of finite dimensional division algebras with centre F forms a group, called the Brauer group Br(F) of F. We will compute the Brauer group of finite fields, the real and complex numbers, p-adic numbers, and most notably of the rational numbers thereby providing a complete classification of division algebras over these fields.

**Books:**

Benson Farb, R. Keith Dennis: *Noncommutative Algebra* (Graduate Texts in Mathematics), ISBN: 038794057X

Richard Pierce: *Associative Algebras*. Graduate Texts in Mathematics, 88. Springer-Verlag, New York-Berlin, ISBN: 0-387-90693-2

Philippe Gille, Tamas Szamuely: *Central Simple Algebras and Galois Cohomology*. Cambridge University Press, Cambridge*,* ISBN: 978-1-316-60988-0