Explicit abelian extensions of complete discrete valuation fields

For higher class field theory Witt and Kummer extensions are very important. In fact, Parshin’s construction of class field theory for higher local fields of prime characteristic [P] is based on an explicit (Artin–Schreier–Witt) pairing; see [F] for a generalization to the case of a perfect residue field. Kummer extensions in the mixed characteristic case can be described by using class field theory and Vostokov’s symbol [V1], [V2]; for a perfect residue field, see [V3], [F]. An explicit description of non Kummer abelian extensions for a complete discrete valuation field K of characteristic 0 with residue field kK of prime characteristic p is an open problem. We are interested in totally ramified extensions, and, therefore, in p-extensions (tame totally ramified abelian extensions are always Kummer and their class field theory can be described by means of the higher tame symbol defined in subsection 6.4.2). In the case of an absolutely unramified K there is a beautiful description of all abelian totally ramified p-extensions in terms of Witt vectors over kK by Kurihara (see section 13 and [K]). Below we give another construction of some totally ramified cyclic p-extensions for such K . The construction is complicated; however, the extensions under consideration are constructed explicitly, and eventually we obtain a certain description of the whole maximal abelian extension of K . Proofs are given in [VZ].