#### 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 $L∕K$ be a field extension of degree $n$. The essential dimension $ed\left(L∕K\right)$ over $k$ is a numerical invariant measuring “the complexity” of $L∕K$. Of particular interest is

also known as the essential dimension of the symmetric group ${S}_{n}$. The exact value of $\tau \left(n\right)$ is known only for $n\le 7$. 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 $n=\left[L:K\right]$ is replaced by a pair $\left(n,e\right)$ which accounts for the size of the separable and the purely inseparable parts of $L∕K$, respectively, and $\tau \left(n\right)$ is replaced by

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 $\tau \left(n,e\right)$.

##### Keywords
inseparable field extension, essential dimension, group scheme in prime characteristic
##### Mathematical Subject Classification 2010
Primary: 12F05, 12F15, 12F20, 20G10