On the properties of the Vostokov and Parshin pairing in higher local class field theory Original Research Article

Let k be a finite extension of Qp which contains the roots of unity μq, where q = pu, p ≠ 2. Vostokov has generalized the norm residue symbol for mixed characteristic n-dimensional local fields Xn = k t1 tn−1. We show here that the Vostokov pairing on Xn commutes with the Parshin pairing defined on its residue field. We also show that the Parshin pairing defined on the n-dimensional local field Xn = Fq((t1))…((tn)) commutes with the Parshin pairing defined on its residue field.