  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).
