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MA4H8 Content

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