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In the present paper, norm series for Honda formal groups are studied. The Steinberg relation for the classical Hilbert symbol is generalized to Honda formal groups. Necessary and sufficient conditions for a series to satisfy the generalized Steinberg relation are obtained. In the paper, such series are called norm series. Bibliography: 8 titles.

In the present paper, necessary and sufficient conditions are given for the equality of the power rezidue symbols $$ {\left( {\frac{\alpha }{a}} \right)_n} $$ and $$ {\left( {\frac{\alpha }{a}} \right)_n} $$ in the cyclotomic field ℚ(ζ n ), 2 ∤ n, for a ∈ ℤ, (a, n) = 1. This result is a generalization of the classical Eisenstein reciprocity law and its continuation in a Hasse’s paper. Bibliography: 3 titles.

In the present paper, the notion of norm series with respect to the norm residue symbol is generalized to high-dimensional local fields. Necessary and sufficient conditions for the existence of norm series are obtained. Bibliography: 12 titles.

The paper is devoted to classification of finite abelian extensions L/K which satisfy the condition [L:K]|D L/K. Here K is a complete discretely valued field of characteristic 0 with an arbitrary residue field of prime characteristic p, D L/K is the different of L/K. This condition means that the depth of ramification in L/K has its “almost maximal” value. The condition appeared to play an important role in the study of additive Galois modules associated with the extension L/K. The study is based on the use of the notion of independently ramified extensions, introduced by the authors. Two principal theorems are proven, describing almost maximally ramified extensions in the cases when the absolute ramification index e is (resp. is not) divisible by p-1. Bibliography: 7 titles.

Let k be the field of formal power series in one variable over a finite field of constants of characteristic p and K/k be a completely ramified extension of degree p. For the group E1 of principal units of the field K, considered as a Galois module, generators are constructed effectively. The operator structure (as a Galois module) of the subgroups Em generating a natural filtration of the group E1 is determined.