Division algebra subfields introduced by an indeterminate
Let \(F\) be the field of fractions of a Henselian discrete rank one valuation ring, and let \(k\) be its residue field. In this paper the structure of subfields of division algebras \(D/F\) is studied. The main case considered is that where \(k\) is a local or global number field. More generally the theory which is developed applies if all division algebras over all finite extensions of \(k\) have period equal to their index (the author calls these fields absolutely stable). It is shown that the isomorphism classes of degree \(n\) totally ramified extensions of \(F\) (denoted with \(R_n(F)\)) admit a natural simple and transitive group action by \(k^*/k^{*n}\). The classes of degree \(n\) totally ramified subfields of division algebras \(D/F\) (notation \(S_n(D)\)) are orbits of naturally defined subgroups. In order to get more information on the existence of totally ramified subfields (of \(D\)) and on totally ramified subfields of a fixed degree, the author introduces invariants \(\underline{\mathbf c}(D)\) and \(\overline{\mathbf c}(D)\). Here \(\underline{\mathbf c}(D)\) is the infimum of the largest degrees of totally ramified subfields of the form \(F(^n\sqrt{ut})\) (where \(u\) is a unit and \(t\) a uniformising element in \(F\)), \(\overline{\mathbf c}(D)\) is the supremum of these numbers. The theory is applied to a series of examples. The classes \(S_n(D)\), the invariants \(\underline{\mathbf c}(D)\) and \(\overline{\mathbf c}(D)\) are computed in the case where the residue field is a local or a global number field.
scientific article; zbMATH DE number 1013922
scientific article