On properties of the n-dimensional norm residue symbol in higher local class field theory Original Research Article

Let k be a finite extension of Qp which contains the roots of unity μ. Here ¦μ¦ = Q = pu, p ≠ 2. We consider an n-dimensional local field given explicitly as a power series in n − 1 variables by Xn = k t1… tn−1. The norm residue symbol has been generalized by Vostokov for mixed characteristic local fields X = t1 tn−1 of dimension n. It is a non-degenerate pairing given by image where Kn(X) is the nth Milnor K-group of X = Xn and X* is the multiplicative group of X. It is shown here that the Vostokov pairing on the n-dimensional local field Xn = k t1 tn−1 commutes with the Vostokov pairing on the n − 1 dimensional local field Xn − 1 = k t1 tn − 2. We achieve this by constructing a map M which projects the roots of unity from the pairing on X = Xn onto the roots of unity from the pairing on Xn − 1.