Brauer p-dimension of complete discretely valued fields.

Let K be a complete discretely valued field of characteristic 0 with residue field \kappa of characteristic p. Let n be the p-rank of κ , i.e., pn = [κ : κp]. It was proved by Parimala and Suresh [Invent. Math. 197 (2014), no. 1, pp. 215-235] 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 period p and index pn+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. [ABSTRACT FROM AUTHOR]