Local differents of algebraic and finite extensions of valued fields (Q1098892)

Let K/k be a finite separable field extension, k henselian for a real valuation v. The author calls \(\Delta (K/k):=[g.c.d\). of the differents of all integral bases of K/k] the algebraic different of K/k, and \(\delta (K/k):=[g.c.d\). of the differents of all integral elements of K/k] the arithmetic different of K/k. In the classical case, where v is discrete and the residue field extension is separable the two notations coincide, and for an intermediate field L one has \(\delta (K/k)=\delta (K/L)\cdot \delta (L/k)\). In general this is not true, even if v is discrete. In this case the author calls K/k ``Dedekindian'' if \(\Delta (K/k)=\delta (K/k)\), and ``Hilbertian'' if the above relation for intermediate fields holds. If v is not discrete he introduces somewhat wider classes of ``quasi-Dedekindian'' and ``quasi-Hilbertian'' field extensions. He proves many theorems about the relations between the two differents and the corresponding discriminants. A decisive tool is the notion of ``semi- real numbers'', which was introduced previously by the author. He also gives a number of (counter-)examples. In a last section connections with ramification theory are studied. (Unfortunately there is a lot of misprints.)