Essential dimension of symmetric groups in prime characteristic

The essential dimension ed(k)(S-n) of the symmetric group Sn is the minimal integer d such that the general polynomial x(n) + a(1)x(n-1) + center dot center dot center dot + a(n) can be reduced to a d-parameter form by a Tschirnhaus transformation. Finding this number is a long-standing open problem, originating in the work of Felix Klein, long before essential dimension of a finite group was formally defined. We now know that ed(k)(S-n) lies between [n/2] and n - 3 for each n >= 5 and any field k of characteristic different from 2. Moreover, if char(k) = 0, then ed(k)(S-n) >= [(n + 1)/2] for any n >= 7. The value of ed(k)(S-n) is not known for any n >= 8 and any field k, though it is widely believed that edk (Sn) should be n - 3 for every n >= 5, at least in characteristic 0. In this paper we show that for every prime p there are infinitely many positive integers n such that ed(Fp) (Sn) <= n - 4.