Mathematical Sciences: Crossed Products over Rational Function Fields

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This project is concerned with problems in ring theory and deals with questions about the maximal subfields of finite dimensional division algebras. The principal investigator will consider the following two questions: (1) given a finite algebraic extension L of a field K, does there exist a finite dimensional division algebra with center K containing L as a maximal subfield?, and (2) given a finite group G which occurs as a Galois group over K, does there exist a Galois extension L of K with the Galois group of L/K isomorphic to G and a factor set f mapping GXG into L* such that the crossed product algebra (L/K,f) is a division ring? The research supported in this project is in the general area of ring theory. The principal investigator will study how one central simple algebra can contain another. This work draws on some deep results in group theory and will shed light on various questions in algebra and number theory.