# 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**: MA3D5 Galois Theory, MA377 Rings and ModulesLink opens in a new window

**Useful background:** helpful but not strictly necessary: p-adic numbers, quadratic forms

**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 some Galois theory and some of MA377 Rings and Modules
- Quaternion algebras and cyclic algebras
- Central simple Algebras and the Brauer group
- Computation of the Brauer group of local fields and of the rational numbers

**Aims**: The goal is to understand one of the major results of 20th century pure mathematics: The Albert-Brauer-Hasse-Noether TheoremLink opens in a new window. The theorem is at the interface of number theory and non-commutative algebra with ramifications to algebraic geometry, representation theory and K-theory; it incapsulates generalisations of various reciprocity laws (quadratic, cubic, etc) and has lead to the modern formulation of class field theory (though we won't have time to go into much of that in the module).

Up to a nilpotent ideal, (possibly non-commutative) Artinian rings, for instance, finite dimensional algebras over a field, are a finite product of matrix rings of division rings. Recall that division rings are those rings in which every non-zero element has a multiplicative inverse. We will learn ways of constructing many examples of division rings. Under a suitable operation, the "set" of finite dimensional division algebras with centre a field F forms a group, called the Brauer group Br(F) of F. We will study and 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. Along the way, we will learn about local fields and quadratic forms which are ubiquitous in mathematics. Module homomorphisms f:M->N applied to an element x of M will be written in the traditional manner as f(x).

**Books: **There will be complete lecture notes for the preparation of which I have used the following 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

Kersten, Ina: Brauergruppen von Körpern.(German) [Brauer groups of fields] Aspects Math., D6, ISBN:3-528-06380-7