Essential dimension of inseparable field extensions

TL;DR

This paper investigates the essential dimension of inseparable field extensions over fields of positive characteristic, providing a simple formula for a generalized invariant that accounts for inseparable parts.

Contribution

It introduces a new framework for measuring the complexity of inseparable extensions using a generalized invariant and computes its value explicitly.

Findings

Derived a simple formula for the essential dimension of inseparable extensions.

Extended the concept of essential dimension to include inseparable parts of field extensions.

Simplified the computation of essential dimension for certain group schemes over fields of positive characteristic.

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 $\tau$(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 $\tau$(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 \tau(n) is replaced by $\tau$(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 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 \tau(n, e).