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(L/K) over k is a numerical invariant measuring "the complexity" of L/K. Of particular interest is τ (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 τ (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 τ (n) is replaced by τ (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 τ (n, e).

From Hilbert’s 13th problem to essential dimension and back

“…Surprisingly, the case where e ≠ Ø (i.e., the polynomials f(x) in question are not separable) turns out to be easier. We refer the reader to [53], where an exact formula for ed(n, e) is obtained.…”