Displaying similar documents to “Generalized Kummer theory and its applications”

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...

Patterns and periodicity in a family of resultants

Kevin G. Hare, David McKinnon, Christopher D. Sinclair (2009)

Journal de Théorie des Nombres de Bordeaux

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Given a monic degree N polynomial f ( x ) ∈ ℤ [ x ] and a non-negative integer ℓ , we may form a new monic degree N polynomial f ℓ ( x ) ∈ ℤ [ x ] by raising each root of f to the ℓ th power. We generalize a lemma of Dobrowolski to show that if m < n and p is prime then p N ( m + 1 ) divides the resultant of f p m and f p n . We then consider the function ( j , k ) ↦ Res ( f j , f k ) mod p m . We show that for fixed p and m that this function is periodic in both j and k , and exhibits high levels of symmetry. Some discussion of its structure as a union of lattices is also given. ...

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...

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 .

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.