Brauer $p$-dimension of complete discretely valued fields

Abstract

Let K be a complete discretely valued field of characteristic 0 with residue field κ of characteristic p. Let n = [κ : κ p ] be the p-rank of κ. It was proved in [PS14] that the Brauer p-dimension of K lies between n/2 and 2n. For n ≤ 3, we improve the upper bound to n + 1 and provide examples to show that our bound is sharp. For n ≤ 2, we also improve the lower bound to n. For general n, we construct a family of fields Kn with residue fields of p-rank n, such that Kn admits a central simple algebra Dn of index p n+1 . Our sharp lower bounds for n ≤ 2 and upper bounds for n ≤ 3 in combination with the nature of these examples motivate us to conjecture that the Brauer p-dimension of such fields always lies between n and n + 1.

Asymptotic Brauer $p$-Dimension

“…This invariant has been studied by many mathematicians. See [10] and [3] for recent results on the subject.…”

“…. , α n ) then there exists a division algebra of exponent p and index p n over F (see [3,Section 5]), hence Brd p (F)…”