A new construction of noncrossed product algebras

Abstract

New examples of noncrossed product division algebras are obtained, using methods different from all previous noncrossed product constructions. The examples are division algebras over intersections of p p -Henselian valued fields, and they have Schur index p m {p^m} and exponent p n {p^n} for any prime number p p and any integers m ⩾ n ⩾ 2 ( n ⩾ 3 if p = 2 ) m \geqslant n \geqslant 2\;(n \geqslant 3\;{\text {if}}\;p = 2) . The basic tools used in the construction are valuation theory and Galois cohomology; no generic methods are applied and there is no p.i. theory. Along the way, local-global principles are proved for central simple algebras over intersections of p p -Henselian valued fields.