Displaying similar documents to “Constructing class fields over local fields”

Absolute norms of p -primary units

Supriya Pisolkar (2009)

Journal de Théorie des Nombres de Bordeaux

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We prove a local analogue of a theorem of J. Martinet about the absolute norm of the relative discriminant ideal of an extension of number fields. The result can be seen as a statement about 2 -primary units. We also prove a similar statement about the absolute norms of p -primary units, for all primes p .

Wintenberger’s functor for abelian extensions

Kevin Keating (2009)

Journal de Théorie des Nombres de Bordeaux

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Let k be a finite field. Wintenberger used the field of norms to give an equivalence between a category whose objects are totally ramified abelian p -adic Lie extensions E / F , where F is a local field with residue field k , and a category whose objects are pairs ( K , A ) , where K ≅ k ( ( T ) ) and A is an abelian p -adic Lie subgroup of Aut k ( K ) . In this paper we extend this equivalence to allow Gal ( E / F ) and A to be arbitrary abelian pro- p groups.

The field-of-norms functor and the Hilbert symbol for higher local fields

Victor Abrashkin, Ruth Jenni (2012)

Journal de Théorie des Nombres de Bordeaux

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The field-of-norms functor is applied to deduce an explicit formula for the Hilbert symbol in the mixed characteristic case from the explicit formula for the Witt symbol in characteristic p > 2 in the context of higher local fields. Is is shown that a “very special case” of this construction gives Vostokov’s explicit formula.

Modified proof of a local analogue of the Grothendieck conjecture

Victor Abrashkin (2010)

Journal de Théorie des Nombres de Bordeaux

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A local analogue of the Grothendieck Conjecture is an equivalence between the category of complete discrete valuation fields K with finite residue fields of characteristic p ≠ 0 and the category of absolute Galois groups of fields K together with their ramification filtrations. The case of characteristic 0 fields K was studied by Mochizuki several years ago. Then the author of this paper proved it by a different method in the case p > 2 (but with no restrictions on the characteristic of K )....