Essential dimension of inseparable field extensions

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

$$\tau \left(n\right)=max\left\{ed\left(L∕K\right)\mid L∕K\text{ is a separable extension of degree }n\right\},$$

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

$$\tau \left(n,e\right)=max\left\{ed\left(L∕K\right)\mid L∕K\text{ is a field extension of type }\left(n,e\right)\right\}.$$

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