Displaying similar documents to “Absolute norms of p -primary units”

Constructing class fields over local fields

Sebastian Pauli (2006)

Journal de Théorie des Nombres de Bordeaux

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Let K be a 𝔭 -adic field. We give an explicit characterization of the abelian extensions of K of degree p by relating the coefficients of the generating polynomials of extensions L / K of degree p to the exponents of generators of the norm group N L / K ( L * ) . This is applied in an algorithm for the construction of class fields of degree p m , which yields an algorithm for the computation of class fields in general.

A classification of the extensions of degree p 2 over ℚ p whose normal closure is a p -extension

Luca Caputo (2007)

Journal de Théorie des Nombres de Bordeaux

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Let k be a finite extension of ℚ p and ℰ k be the set of the extensions of degree p 2 over k whose normal closure is a p -extension. For a fixed discriminant, we show how many extensions there are in ℰ ℚ p with such discriminant, and we give the discriminant and the Galois group (together with its filtration of the ramification groups) of their normal closure. We show how this method can be generalized to get a classification of the extensions in ℰ k .

Hilbert-Speiser number fields and Stickelberger ideals

Humio Ichimura (2009)

Journal de Théorie des Nombres de Bordeaux

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Let p be a prime number. We say that a number field F satisfies the condition ( H p n ′ ) when any abelian extension N / F of exponent dividing p n has a normal integral basis with respect to the ring of p -integers. We also say that F satisfies ( H p ∞ ′ ) when it satisfies ( H p n ′ ) for all n ≥ 1 . It is known that the rationals ℚ satisfy ( H p ∞ ′ ) for all prime numbers p . In this paper, we give a simple condition for a number field F to satisfy ( H p n ′ ) in terms of the ideal class group of K = F ( ζ p n ) and a “Stickelberger ideal” associated to the...

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.

Conjugacy classes of series in positive characteristic and Witt vectors.

Sandrine Jean (2009)

Journal de Théorie des Nombres de Bordeaux

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Let k be the algebraic closure of 𝔽 p and K be the local field of formal power series with coefficients in k . The aim of this paper is the description of the set 𝒴 n of conjugacy classes of series of order p n for the composition law. This work is concerned with the formal power series with coefficients in a field of characteristic p which are invertible and of finite order p n for the composition law. In order to investigate Oort’s conjecture, I give a description of conjugacy classes of series...