Vol. 13, No. 2, 2019

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Essential dimension of inseparable field extensions

Zinovy Reichstein and Abhishek Kumar Shukla

Vol. 13 (2019), No. 2, 513–530
Abstract

Let k be a base field, K be a field containing k, and LK be a field extension of degree n. The essential dimension ed(LK) over k is a numerical invariant measuring “the complexity” of LK. Of particular interest is

τ(n) = max{ed(LK) LK is a separable extension of degree n},

also known as the essential dimension of the symmetric group Sn. The exact value of τ(n) is known only for n 7. In this paper we assume that k is a field of characteristic p > 0 and study the essential dimension of inseparable extensions LK. Here the degree n = [L : K] is replaced by a pair (n,e) which accounts for the size of the separable and the purely inseparable parts of LK, respectively, and τ(n) is replaced by

τ(n,e) = max{ed(LK) LK is a field extension of type (n,e)}.

The symmetric group Sn is replaced by a certain group scheme Gn,e over k. This group scheme is neither finite nor smooth; nevertheless, computing its essential dimension turns out to be easier than computing the essential dimension of Sn. Our main result is a simple formula for τ(n,e).

Keywords
inseparable field extension, essential dimension, group scheme in prime characteristic
Mathematical Subject Classification 2010
Primary: 12F05, 12F15, 12F20, 20G10
Milestones
Received: 28 June 2018
Accepted: 24 December 2018
Published: 2 March 2019
Authors
Zinovy Reichstein
Department of Mathematics
University of British Columbia
Vancouver, BC
Canada
Abhishek Kumar Shukla
Department of Mathematics
University of British Columbia
Vancouver, BC
Canada