Essential dimension of inseparable field extensions

Reichstein, Zinovy; Shukla, Abhishek

doi:10.2140/ant.2019.13.513

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Essential dimension of inseparable field extensions

Zinovy Reichstein and Abhishek Kumar Shukla

Vol. 13 (2019), No. 2, 513–530

DOI: 10.2140/ant.2019.13.513

Abstract

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

τ (n) = max {ed (L ∕ K) ∣ L ∕ K is a separable extension of degree n},

also known as the essential dimension of the symmetric group $S_{n}$ . The exact value of $τ (n)$ is known only for $n \leq 7$ . In this paper we assume that $k$ is a field of characteristic $p > 0$ and study the essential dimension of inseparable extensions $L ∕ K$ . 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 $L ∕ K$ , respectively, and $τ (n)$ is replaced by

τ (n, e) = max {ed (L ∕ K) ∣ L ∕ K is a field extension of type (n, e)} .

The symmetric group $S_{n}$ is replaced by a certain group scheme $G_{n, e}$ 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 $S_{n}$ . Our main result is a simple formula for $τ (n, e)$ .

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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

Vol. 13, No. 2, 2019