The Hilbert pairing for formal groups over $\sigma$-rings

In the paper formal groups over the rings of integers of $\sigma$-fields are studied. These fields were constructed by the first-named author in the preceding paper. They are a generalisation of the inertia field of a classical local field to arbitrary complete discrete valuation field of characteristic zero. An analogue of Honda's theory for such formal groups is constructed. The arithmetic of the group of points in an extension of a $\sigma$-field that contains sufficiently many torsion points is studied. Using the classification of formal groups and the arithmetic results obtained an explicit formula for the Hilbert pairing for formal groups over $\sigma$-fields is proved.