A finite dimensional division algebra is called a crossed product if it contains a maximal subfield which is Galois over its center, otherwise a noncrossed product.

Since Amitsur settled the long standing open problem of existence of noncrossed products, their existence over familiar fields was an object of investigation. The simplest fields over which they occur are Henselian fields with global residue field (such as Q((x)), where Q is the field of rational numbers). We shall describe the "location" of noncrossed products over such fields by proving the existence of bounds that, roughly speaking, separate crossed and noncrossed products. Furthermore, we describe those bounds in terms of Grunwald-Wang type of problems and address their solvability in various cases.

(joint work with Timo Hanke and Jack Sonn)