Brauer p-dimension of complete discretely valued fields

TL;DR

This paper investigates the Brauer p-dimension of complete discretely valued fields with residue fields of characteristic p, providing improved bounds and conjecturing a tight range based on p-rank.

Contribution

It improves known bounds on the Brauer p-dimension for certain p-ranks and constructs explicit examples supporting a conjecture about its precise range.

Findings

For n<4, upper bound improved to n+1.

For n<3, lower bound improved to n.

Constructed fields with index p^{n+1} for residue p-rank n.

Abstract

Let K be a complete discretely valued field of characteristic 0 with residue field k of characteristic p. Let n=[k:k^p] be the p-rank of k. It was proved by Parimala and Suresh that the Brauer p-dimension of K lies between n/2 and 2n. For n< 4, we improve the upper bound to n+1 and provide examples to show that our bound is sharp. For n < 3, we also improve the lower bound to n. For general $n$, we construct a family of fields K_n with residue fields of p-rank n, such that K_n admits a central simple algebra D_n of index p^{n+1}. Our sharp lower bounds for n<3 and upper bounds for n< 4 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.