On the norm property of the Hilbert symbol for polynomial formal modules in a multidimensional local field

Abstract

In a two-dimensional local field K containing the pth root of unity, a polynomial formal group F c (X, Y) = X + Y + cXY acting on the maximal ideal M of the ring of integers б K and a constructive Hilbert pairing {·, ·} c : K 2(K) × F c (M) → c , where c is the module of roots of [p] c (pth degree isogeny of F c ) with respect to formal summation are considered. For the extension of two-dimensional local fields L/K, a norm map of Milnor groups Norm: K 2(L) → K 2(K) is considered. Its images are called norms in K 2(L). The main finding of this study is that the norm property of pairing {·, ·}c: {x,β} c : = 0 ⇔ x is a norm in K 2(K([p] c -1 (β))), where [p] c -1 (β) are the roots of the equation [p] c = β, is checked constructively.