Semiramified division algebras over Henselian fields

(eng) The present work studies the valuative structure of some extensively used division algebras in the theory of central simple algebras. A first example of such division algebras is the generic abelian crossed product which served to settle many interesting conjectures concerning the cyclicity of division p-algebras, the existence of division algebras with involution of the first kind which are not tensor products of quaternion algebras and the existence of indecomposable division p-algebras of prime exponent. We show in the first chapter that these generic abelian crossed products are algebras of central quotients of semiramified graded division algebras over graded fields. This new structure enables us to generalize many constructions previously made with generic abelian crossed products. Also, it allows us to classify semiramified graded division algebras that have the same support and the same 0-component. In the second chapter, we study graded field extensions and give many new results concerning especially tame and normal graded field extensions. We study then nicely semiramified division algebras over Henselian fields and simplify the necessary and sufficient conditions for division algebras over a Henselian field to be nicely semiramified. We define what we call nondegenerate division algebras over Henselian fields and use them to give examples of indecomposable division algebras of prime exponent. Also, we describe the valuative structure of nondegenerate tame semiramified division algebras over Henselian fields. Finally, in the last chapter, we determine necessary and sufficient conditions for tame division algebras over a Henselian field to have Kummer subfields. This generalizes the study of Amitsur and Tignol for Malcev-Neumann division algebras and also the study of Morandi and Sethuraman for a special kind of tame division algebras over Henselian fields. We give also conditions for a tame semiramified division algebras over a Henselian field to by non-cyclic and also to be a non elementary abelian crossed product.