The structure of GALols groups of local fields has been studied by many mathematicians.
The description of maximal p-extensions was obtained by 1. R . SAFAREVIC [S] and S . P. DEMUSHKIN [D]. Important results about the GALOIS group of an algebraic closure of local fields were proved by K. IWASAWA [I] and H. KOCH [Kl], [K3]. In 1968 A. V. YAKOVLEV [Y] determined completely the structure of this group. Recently U. JANNSEN and K. WINGBERG obtained a simpler description based on the notion of DEMUSHKIN formation (see [JJ, [JW], [W]).
In this paper we use the generalised HILBERT pairing for the investigation of p-extensions of multidimensional local fields, introduced by A. N. PARSHIN [P 11 and K. KATO [Ka]. We assume that F is an n-dimensional local field of characteristic 0 with a first residue field of characteristic p > 2. Let F contain a primitive q = p"-th root of unity. Theorem 1.1 and Proposition 2.2 describe the structure of the GALOIS group G, of a maximal p-extension of F modulo the third step in the descending q-central series of G,.
This description is very complicated, so in S; 3 we give some explicit calculations for the simplest case
I. The norm pairing for multidimensional local fields
Let F be a complete discrete valuation field. F is said to be an n-dimensional localfield if there exists the sequence k"), k('), . . . , k("-), A(") = F of complete discrete valuation fields such that k") = Fqo is finite and k'"' is a residue field of k"' ') for each i 5 n -1. For example, let K be a complete discrete valuation field. Denote by K ( { t } } the field of all power series 1 a i f i such that a, + 0 if i -,co. If K is an ordinary local field, then
where k is a residue field of K (see [P 13). A. N. PARSHIN [P 21 and K. KATO [Ka] created multidimensional class fields theory. In this paper we assume that char F = 0 and char k ( " -' ) = p > 2. Let the group pq of