Explicit construction of class field theory for a~multidimensional local field

S. V. Vostokov

Izvestiya. Mathematics , Tome 26 (1986) no. 2, pp. 263-287

Résumé

Let $k$ be a finite extension of the field of $p$-adic numbers $\mathbf Q_p$ and $k\{\{t\}\}$ the field of Laurent series $\sum_{-\infty}^\infty a_it^i$ for which the $a_i$ are bounded in the norm of $k$ and $a_i\to0$ as $i\to-\infty$. In the $n$-dimensional local field $F=k\{\{t_1\}\}\cdots\{\{t_{n-1}\}\}$ a pairing is given in explicit form between the completed Milnor $k$-functor $K_n^{\mathrm{top}}(F)$ and the multiplicative group $F^*$ with values in the group of $q=p^m$th roots of unity. Bibliography: 14 titles.