On ^-regular Extensions of Local Fields1

1. Let A be a complete field with respect to a discreet valuation,2 and suppose that the residue class field of K is finite and has characteristic p. A group which is finite and whose order is not divisible by p is said to be ^-regular. A normal extension of A whose Galois group is ^-regular will be called a ^-regular extension of K. The object of this paper is to characterize those groups which are Galois groups of ^-regular extensions of K, and to give a criterion for deciding how many ^-regular extensions of A have a given group as Galois group. It will be necessary to make use of a result closely related to a theorem of Iwasawa [2, Theorem 2, p. 463].