Generic abelian crossed products and graded division algebras

Subfields of Nondegenerate Tame Semiramified Division Algebras

“…In particular, if L is a Galois maximal subfield of A, then Gal(L/Z) ∼ = G.Proof. As recalled preceding Remark 2.1 above, by[BM, Th. 1.1], A = q(B) for some semiramified graded division algebra B with B 0 = N , Z(B) 0 = E,and q(Z(B)) = Z.…”

“…Let F = Z(B), a graded field. It was shown further in [BM,Th. 1.1] that q(F ) = Z, F 0 = E, and B is inertially split and semiramified over F with B 0 = N and Γ B = Z m .…”

“…A is completely determined by N , G, the u ij , and the b i , so we write A = (N/E, G, S, U, b), where U = (u i,j ) 1≤i,j≤m , b = (b i ) 1≤i≤m , and S = (σ i ) 1≤i≤m is the chosen base of G. Amitsur and Saltman defined in [AS,p. 81] a condition that they called degeneracy for the matrix (u i,j ), which by [BM,Prop. 0.13] is equivalent to: there is a field L, E ⊆ L ⊆ N such that Gal(N/L) is noncyclic and [C A (L)] ∈ Dec(N/L).…”

“…2.3] that M is degenerate in A ′ iff N is degenerate in A. A ′ is definable as the ring of quotients of an iterated twisted polynomial ring, and it was shown in [BM,Th. 1.1] that A ′ is therefore also q(B) for a graded division ring B, which is an iterated twisted Laurent polynomial ring.…”

Prime to p extensions of the generic abelian crossed product

“…This theorem compares to [BM00, Lemma 3.1 and Theorem 3.2]. In [BM00], Boulagouaz and Mounirh use semi-ramified, central graded division algebras to get their result on prime power central elements. It will be convenient for us to have the result over valued fields because we will apply (2.12) to extensions of Henselian valued fields.…”