Generalized Kummer theory and its applications

Toru Komatsu

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 $(H_{p^{\infty}}^{'})$ when it satisfies $(H_{p^{n}}^{'})$ for all $n \geq 1$ . It is known that the rationals $ℚ$ satisfy $(H_{p^{\infty}}^{'})$ 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 $f (x) \in ℤ [x]$ and a non-negative integer $ℓ$ , we may form a new monic degree $N$ polynomial $f_{ℓ} (x) \in ℤ [x]$ 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 $(j, k) \mapsto Res (f_{j}, f_{k}) mod p^{m}$ . 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.

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

Hilbert-Speiser number fields and Stickelberger ideals

Patterns and periodicity in a family of resultants

Conjugacy classes of series in positive characteristic and Witt vectors.

Absolute norms of p -primary units

Constructing class fields over local fields

Absolute norms of $p$ -primary units