NSF proposal: ``Division algebras and field invariants'' P.I.: David J. Saltman
Fractions of integers, which we all learned about in the early grades, form a system called a ``field''. But this field is not large enough, since in this so called ``rational'' field you cannot take the square root of 2, or write down pi. It has become, therefore, a truism that we need to study all fields. But even the study of all fields is not encompassing enough, because in physics, or when using matrices, one finds one must study objects which are field-like, but which are noncommutative, meaning A times B might not equal B times A. Such objects are called division algebras. The question, in the largest sense, that this grant considers is the classification of all division algebras in terms of the better understood fields, with particular reference to the ``center'' of the division algebra, which is inside the division algebra and is a field. Coupled with the center is the ``degree'' of a division algebra, which is a positive integer that measures how much bigger the division algebras is as compared to its center. Division algebras are an old subject, with a great deal of information known when the center is one dimensional. For example when the center is close to the rational field, any division algebra is known to be so called ``cyclic'', which means there is a very good description in terms of fields. The focus of this proposal is the study of division algebras whose centers have dimension 2. For example, a major focus is the case where the center comes from a curve over a p-adic field, a very special case of a 2 dimensional field. In this case, not all division algebras are cyclic, but the proposers hopes to use algebraic geometry to show that when the division algebra has degree a prime integer, and the center comes from a p-adic curve, then cyclicity does hold.
Perhaps the most important tool in studying division algebras is Galois cohomology, because the set of all division algebras with fixed center F form a group isomorphic to the second Galois cohomology group with unit coefficients. Since second cohomology can be hard to compute, often one attacks this group by using ramification. In more detail, any discrete valuation defines a map to a first cohomology group, and by using all possible discrete valuations one hopes to capture the division algebras. In special cases this is known to work, for example for rational field or the fields arising from p-adic curves. In both cases, one shows, or hopes to show, cyclicity by showing there is a cyclic Galois extension which splits all the ramification of the division algebra. This leads to a more fundamental question. If D is a division algebra of prime degree, is there always a cyclic field extension, of the same degree, which at least splits all the ramification?
