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: K₂(K) × F_c(M) → <ξ>_c, where <ξ>_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: K₂(L) → K₂(K) is considered. Its images are called norms in K₂(L). The main finding of this study is that the norm property of pairing {·, ·}c: {x,β}_c: = 0 ⇔ x is a norm in K₂(K([p]_c^-1(β))), where [p]_c^-1(β) are the roots of the equation [p]_c = β, is checked constructively.