Galois subfields of division algebras
Daniel Krashen, Penn
Abstract: In this talk I will discuss the problem of characterizing
the possible groups which may occur as Galois groups of a maximal
subfield of a division algebra. This question depends strongly on the
arithmetic of the center of the algebra, and is unknown in a great
number of cases, even in the case where the center is the rational
numbers. In joint work with Harbater and Hartmann, we have completely
answered this question when the center is a quasi-global field (the
function field of a curve over a comple discretely valued field with
algebraically closed residue field of characteristic 0). The proof
makes use of the technique of "field patching," developed by Harbater
and Hartmann.
This problem may also be interpreted purely in terms of finding
rational points on certain varieties after a field extension of
"minimal size", and I will also describe this point of view.