Kummer subfields of tame division algebras over Henselian fields

Abstract

Communicated by E.M. Friedlander MSC: 16K50 16W50 16W60 16W70 a b s t r a c t By generalizing the method used by Tignol and Amitsur in [J.-P. Tignol, S.A. Amitsur, Kummer subfields of Malcev-Neumann division algebras, Israel Journal of Math. 50 (1985), 114-144], we determine necessary and sufficient conditions for an arbitrary central division algebra D over a Henselian valued field E to have Kummer subfields when the characteristic of the residue field E of E does not divide the degree of D. We prove also that if D is a semiramified division algebra of degree n [resp., of prime power degree p r ] over E such that char(E) does not divide n and rk(Γ D /Γ E ) ≥ 3 [resp., p = char(E) and p 3 divides exp(Γ D /Γ E )], then D is non-cyclic [resp., D is not an elementary abelian crossed product].

Discriminants of orthogonal involutions on central simple algebras with tame gauges

“…We will show in this article that if A is a graded simple algebra with simple 0-component, then A is a generalized graded crossed product (see Lemma 2.4). This lemma can be used to generalize many results in [11]. It will also allow us to show that if A is an inertially split graded simple algebra with A 0 simple, and if A has a graded involution of the first kind σ with restriction σ |A 0 to A 0 also of the first kind, then σ and σ |A 0 have the same type (see Proposition 2.8 and Lemma 2.12).…”

“…These algebras were used by Jacob and Wadsworth in [9] to study inertially split division algebras over a Henselian valued field, and recently they were used by the author in [11] to give necessary and sufficient conditions for the existence of Kummer subfields in central division algebras over Henselian fields of residue characteristic not dividing the degree. We will show in this article that if A is a graded simple algebra with simple 0-component, then A is a generalized graded crossed product (see Lemma 2.4).…”

“…In the same way as above, if ω : H → Aut(A) 0 and f : H × H → A * are two maps that satisfy the conditions (1)-(3) above, we say that (ω, f ) is a graded factor set of H in A. The corresponding generalized graded crossed product (A, H, (ω, f )) is defined also in the same way: (A, H, (ω, f )) = ⊕ σ ∈H Ax σ , where x σ are independent indeterminates on A satisfying the multiplicative conditions x σ a = ω σ (a)x σ and x σ x τ = f (σ , τ )x σ τ for all a ∈ A and σ , τ ∈ H. We recall that there is a unique graded algebra structure on (A, H, (ω, f )) which extends the grading of A and for which x σ are homogeneous elements [11,Lemma 1.2].…”

“…(2) We recall that the author gave in [11] necessary and sufficient conditions for the existence of Kummer subfields in an arbitrary central division algebra D over a Henselian valued field when the residue characteristic does not divide deg(D). A main tool that we used in [11] was the fact that GD can be written as a generalized graded crossed product.…”

“…A main tool that we used in [11] was the fact that GD can be written as a generalized graded crossed product. The reader can see that using Lemma 2.4 we can prove many results of [11] in the more general context of a graded central simple algebra A with simple 0-component A 0 .…”