Essential dimension of finite p-groups

We prove that the essential dimension of a p-group G over a eld F containing a primitive p-th root of unity is equal to the least dimension of a faithful representation of G over F. The notion of the essential dimension edF (G) of a nite group G over a eld F was introduced in (5). The integer edF (G) is equal to the smallest number of algebraically independent parameters needed to classify all Galois G-algebras over any eld extension of F. If V is a faithful linear representation of G over F , then edF (G) dim(V ) (cf. (2, Prop. 4.11)). The essential dimension of G can be smaller than dim(V ) for every faithful representation V of G over F. For example, we have edF (Z=3Z) = 1 over F = Q or any eld F of characteristic 3 (cf. (12), (2, Example 2.3)) and edC(S5) = 2 (cf. (5, Th. 6.5)). In this paper we prove that if G is a p-group and F is a eld of characteristic dieren t from p containing a primitive p-th root of unity, then edF (G) coincides with the least dimension of a faithful representation of G over F. In the paper the word \scheme" means a separated scheme of nite type over a eld and \variety" an integral scheme. Acknowledgment: We are grateful to Z. Reichstein for useful conversations.