Since Amitsur settled the long standing fundamental open problem of existence of noncrossed products, their existence over familiar fields was an object of investigation. In the first talk, we describe a fundamental valuation theoretic argument (original to Eric Brussel) which led to the discovery of noncrossed products over Henselian valued fields F with global residue field. In the second talk, we shall describe the "location" of noncrossed products in the Brauer group of F by proving the existence of certain bounds on the index that, roughly speaking, separate crossed and noncrossed products. (Joint with Jack Sonn.)

Since Amitsur settled the long standing fundamental open problem of existence of noncrossed products, their existence over familiar fields was an object of investigation. In the first talk, we describe a fundamental valuation theoretic argument (original to Eric Brussel) which led to the discovery of noncrossed products over Henselian valued fields F with global residue field. In the second talk, we shall describe the "location" of noncrossed products in the Brauer group of F by proving the existence of certain bounds on the index that, roughly speaking, separate crossed and noncrossed products. (Joint with Jack Sonn.)