Essential dimension of inseparable field extensions

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) vertical bar 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 <= 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) vertical bar 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 tau(n, e).