Local differents of algebraic and finite extensions of valued fields

Abstract

The properties of discriminants and differents were studied first by Dedekind and Hilbert in finite algebraic extensions of fields of algebraic numbers. From a local point of view, that is equivalent to a study of the p-adic case, where the results of Dedekind and Hilbert can be formulated as follows. Dedekind's theorem: The g.c.d. Δ( K k ) of differents of integral bases of a finite algebraic extension K k (which I call an algebraic different if K k and the g.c.d δ( K k ) of differents of integral elements of K k (which I call an arithmetic different of K k ) coincide; Hilbert's theorem (which is the basis of Herbrand's ramification theory of intermediate extensions): If K ⊃ L ⊃ k, δ( K k ) = δ( K L ) δ( L k ) . These results are easily generalizable to the “classical case” of henselian valued basic fields, i.e., the case when the valuation is discrete and the residual extension K k of K k is separable. But, in the general case of extensions K k of valued fields (where k may be assumed to be henselian), Dedekind's and Hilbert's theorems are not always true: the algebraic different Δ( K k ) divides the arithmetic different δ( K k ) , but generally δ( K k ) ≠ Δ( K k ) , and Hilbert's theorem holds only for the algebraic different. When the valuation is discrete, I call an extension K k dedekindian when δ( K k ) = Δ( K k ) and hilbertian if, for every intermediate field L of K k (i.e., K ⊇ L ⊇ k), Hilbert's theorem δ( K k ) = δ( K L ) δ( L k ) for arithmetic differents holds. When the valuation is dense, the situation is more complicated, because of the existence of two kinds of ideals (principal and other), and it is convenient to define dedekindian and hilbertian extensions in a slightly different manner and to introduce somewhat wider classes of extensions called quasi-dedekindian and quasi-hilbertian. I study the relations between Δ( K k ) and δ( K k ) , and, in particular, I give a complete characterization of dedekindian extensions for both discrete and dense valuations; I also give examples of non-dedekindian and non-hilbertian extensions. In Section 4, some connections with the ramification theory (both for normal and non-normal extensions) are studied and a weak analog of Hilbert's theorem [ δ( K k ) δ( L k ) divides δ( K k ) ] is proved.