RÉSUMÉ

In 1972 Epp proved a theorem asserting that in an arbitrary finite extension of discretely valued fields L/K one can eliminate wild ramification, i.e. to guarantee validity of e(k'L/k'K)=1 for some finite extension k'/k, where k is a constant subfield: maximal subfield of K with perfect residue field (k is canonical in the mixed characteristic case). We prove a variant of Epp's theorem, which asserts that for k'/k one can take a composite of a cyclic extension and an extension of some bounded degree.

Moreover, the cyclic extension can be chosen inside any given deeply ramified extension of k, in particular, inside any ramified Z_p-extension.

The proof proceeds for the equal and mixed characteristic cases in parallel. It is based on the splitting of an arbitarary finite extension into a tower of an "almost constant" and "infernal" extensions. (An extension is said to be almost constant if it can be embedded into a composite of a constant extension and absolutely unramified one. An extension is said to be infernal if it possesses no non-trivial almost constant subextension.) We remark that elimination of wild ramification in an infernal extension yields a ferociously ramified extension of the same degree, that means an extension of ramification index 1 and purely inseparable extension of the residue fields.

As an application, we deduce a new version of classification theorem for higher-dimensional local fields.

Next, we show that in the equal characteristic case by means of a suitable choice of the constant subfield one can force any given Galois p-extension to be infernal. This permits one to eliminate ramification by means of a cyclic k'/k (under some obvious restrictions).

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Dernière mise à jour le 27 février 1998.