Mathematical Sciences: Brauer Groups and the Theory of Fields

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Saltman This award supports research on both Brauer groups and the rationality problem for invariant fields. The principal investigator is specifically interested in the rationality problem for the center of the generic division algebra. A considerable part of this proposal concerns all these subjects, linked by the unramified cohomology. The principal investigator will determine if the unramified cohomology can be used to show the center of the generic division algebra is nonrational and if it can be used to show other invariant fields are nonrational. He will also see if the Merkuriev-Suslin theorem can be made more constructible and if the theory of toric varieties can be used to find good projective models for multiplicative invariant fields. The research supported is in the general area of field theory. Fields are algebraic structures similar to the rational numbers, the real numbers, and the complex numbers in that each nonzero element has an inverse with respect to multiplication. Field theory is a rich and deep subject with its historical roots arising from the early attempts to systematically describe the zeros of polynomials. ***