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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

Additional Resources

Archived Pages: 2016 2017 2018