Explicit formulas for the Hilbert symbol. (English) Zbl 1008.11053

Vostokov, Sergei V.

Fesenko, Ivan (ed.) et al., Invitation to higher local fields. Extended version of talks given at the conference on higher local fields, Münster, Germany, August 29-September 5, 1999. Coventry: Geometry and Topology Publications, Geom. Topol. Monogr. 3, 81-89 (2000).

This is a review of the vast area of explicit formulas for the (wild) Hilbert symbol (not only in the one-dimensional case but in the higher dimensional case as well). An extensive bibliography is included.
The first section reviews Kummer type formulas (Shafarevich (1950), S. Vostokov (1978), H. Brückner (1979)) and the second one review Artin-Hasse type formulas (K. Iwasawa (1968), S. Sen (1980)). Then connections with class field theory, \(p\)-divisible groups, \(p\)-adic representations are given. The final section gives explicit formulas in higher-dimensional fields of characteristic 0.
For the entire collection see [Zbl 0954.00026].

Reviewer: O.Ninnemann (Berlin)

MSC:

11S31	Class field theory; \(p\)-adic formal groups
11S70	\(K\)-theory of local fields

Keywords:

explicit formulas for the Hilbert symbol; arithmetic