𝑝-algebras over maximally complete fields

On wild division algebras over fields of power series

“…6. of Tignol in Appendix to the paper [2]. These lemmas show that a tensor product A 1 ⊗ A 2 of any two symbols A 1 , A 2 is similar either to a single symbol in Br(F ) (in which case we are done) or to a product of two symbols of level zero.…”

“…This conjecture was proposed by M. Artin and was solved for some another examples of the field F by many authors. As it is known for me, the positive answer for all division algebras of index indA = 2 a 3 b was given in [10], for division algebras over the field F = k((X))((Y )) , where k is a perfect field of characteristic p = 0 such that dim Fp k/℘(k) = 1 , was given by Tignol in the Appendix in [2] (we include this case though F may not be a C 2 -field), for division algebras of index prime to the characterictic of F , where F is a function field of a surface, was given in [7]. I propose, the positive answer was also known for division algebras over F = F 1 ((t)) of characteristic 0.…”

Noncyclic division algebras over fields of Brauer dimension one

“…Recent work on Brauer dimension includes [3], [5], [9], [10], [11], [12], [16], [17], [20], [21], and [22]. Two-dimensional local fields, such as Qp((t)) and Fp((s))((t)), were shown to have Brauer dimension one in [9], and Fp((s))((t)) was shown to have cyclic length one in [5]. The Brauer dimension of the function field of a p-adic curve was proved to be two in [22], and shown to have cyclic length two in [8].…”

Linkage of sets of Quaternion Algebras in characteristic 2

“…We say a field F is m-linked if any m quaternion algebras over F are linked. Examples of 2-linked fields that are not 3-linked exist: F 2 ((α))((β)) is 2-linked by [AJ95] but not 3-linked by [CDL18]. The question of whether 3-linked fields which are not 4linked exist was raised in [Bec18].…”