On the properties of the Vostokov and Parshin pairing in higher local class field theory (Q1917379)

Let \(k\) be a finite extension of \(\mathbb{Q}_p\) containing the roots of unity of order \(q\), a power of an odd prime \(p\). Vostokov defined the norm residue symbol for \(n\)-dimensional local fields \(X_n= k\{\{t_1\}\} \cdots \{\{t_{n-1}\}\}\) by a pairing. The present author shows that the Vostokov pairing commutes with the Parshin pairing defined on the residue field [see also the preceding review].