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1. The theorems of local class field theory are known to hold for all fields which are complete under a discrete rank one valuation and whose residue class fields (1) have no inseparable extension and (2) have, in any algebraic closure, exactly one (necessarily cyclic) extension of degree n for every integer n> 0. Since complete fields with any given residue class field can be constructed by formal power series or Witt vectors, we shall restrict ourselves to the set of fields satisfying (1) and (2), calling them quasi Galois fields (qGf). 0. F. G. Schilling [3 ] has constructed qGf of characteristic 0 by use of formal power series. But the only qGf's of prime characteristic mentioned up to now are the Galois fields and those infinite algebraic extensions of them whose "degree" has no infinite part. It is easy to see that these are the only absolutely algebraic qGf of prime characteristic. If these should be the only qGf of prime characteristic it would mean that generalized local class field theory was only a limiting case of the classical theory so that the new methods used to prove the existence theorem [5; 6] could be replaced by something much simpler. We prove here that this is not the case. For example, there exist qGf whose absolutely algebraic subfield is the algebraic closure of the Galois field of order p. In fact we prove a much more general result, namely:

The generality of local class field theory (Generalized local class field theory. V.)