Abstract. New sufficient conditions for the validity of local class field theory for Henselian valued fields are established. An example is presented to show that these conditions are less restrictive than the applicability of the Neukirch abstract class field theory. §0. IntroductionLocal class field theory is treated in many papers, monographs, and textbooks (see, e.g., the bibliography in [1]). In the present paper, we suggest yet another exposition based on abstract class field theory [2]. (In the author's paper [3], the reader will find a modified presentation of abstract class field theory on the basis of a combination of the approaches by Neukirch and Hasewinkel.)We recall some definitions and notation. Let F be a field; a subring R of F is called a valuation ring of F if for every element a ∈ F × F \ {0} we have a ∈ R or a −1 ∈ R. Every valuation ring R is local, i.e., has a unique maximal ideal, which will be denoted by m(R); the field F R R/m(R) is called the residue field of R. We denote by U R the multiplicative group of units (invertible elements) of R, and by Γ R the factor group F × /U R ; the group Γ R admits a natural linear order, which is determined by the cone R × /U R ⊆ Γ R . The mapping v R : a → aU R , a ∈ F × , from F × to Γ R is called the valuation of F determined by the valuation ring R. Let U