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A determinant formula for the relative class number of an imaginary abelian number field

Mikihito Hirabayashi (2014)

Communications in Mathematics

We give a new formula for the relative class number of an imaginary abelian number field K by means of determinant with elements being integers of a cyclotomic field generated by the values of an odd Dirichlet character associated to K . We prove it by a specialization of determinant formula of Hasse.

A generalization of a result on integers in metacyclic extensions

James Carter (1999)

Colloquium Mathematicae

Let p be an odd prime and let c be an integer such that c>1 and c divides p-1. Let G be a metacyclic group of order pc and let k be a field such that pc is prime to the characteristic of k. Assume that k contains a primitive pcth root of unity. We first characterize the normal extensions L/k with Galois group isomorphic to G when p and c satisfy a certain condition. Then we apply our characterization to the case in which k is an algebraic number field with ring of integers ℴ, and, assuming some...

A note on Sinnott's index formula

Kazuhiro Dohmae (1997)

Acta Arithmetica

Let k be an (imaginary or real) abelian number field whose conductor has two distinct prime divisors. We shall construct a basis for the group C of circular units in k and compute the index of C in the group E of units in k. This result is a generalization of Theorem 3.3 in a previous paper [1].

Anneaux d’entiers stablement libres sur [ H 8 × C 2 ]

Jean Cougnard (1998)

Journal de théorie des nombres de Bordeaux

Le groupe H 8 × C 2 est le plus petit groupe pour lequel existent des modules stablement libres non libres. On montre que toutes les classes d’isomorphisme de tels modules peuvent être représentées une infinité de fois par des anneaux d’entiers. On applique un travail de classification de Swan, pour cela on doit construire explicitement des bases normales d’entiers d’extensions à groupe H 8 ; cela se fait en liant un critère de Martinet avec une construction de Witt.

Annihilators of minus class groups of imaginary abelian fields

Cornelius Greither, Radan Kučera (2007)

Annales de l’institut Fourier

For certain imaginary abelian fields we find annihilators of the minus part of the class group outside the Stickelberger ideal. Depending on the exact situation, we use different techniques to do this. Our theoretical results are complemented by numerical calculations concerning borderline cases.

Annihilators of the class group of a compositum of quadratic fields

Jan Herman (2013)

Archivum Mathematicum

This paper is devoted to a construction of new annihilators of the ideal class group of a tamely ramified compositum of quadratic fields. These annihilators are produced by a modified Rubin’s machinery. The aim of this modification is to give a stronger annihilation statement for this specific type of fields.

Capitulation and transfer kernels

K. W. Gruenberg, A. Weiss (2000)

Journal de théorie des nombres de Bordeaux

If K / k is a finite Galois extension of number fields with Galois group G , then the kernel of the capitulation map C l k C l K of ideal class groups is isomorphic to the kernel X ( H ) of the transfer map H / H ' A , where H = Gal ( K ˜ / k ) , A = Gal ( K ˜ / K ) and K ˜ is the Hilbert class field of K . H. Suzuki proved that when G is abelian, | G | divides | X ( H ) | . We call a finite abelian group X a transfer kernel for G if X X ( H ) for some group extension A H G . After characterizing transfer kernels in terms of integral representations of G , we show that X is a transfer kernel for...

Circular units of real abelian fields with four ramified primes

Vladimír Sedláček (2017)

Archivum Mathematicum

In this paper we study the groups of circular numbers and circular units in Sinnott’s sense in real abelian fields with exactly four ramified primes under certain conditions. More specifically, we construct -bases for them in five special infinite families of cases. We also derive some results about the corresponding module of relations (in one family of cases, we show that the module of Ennola relations is cyclic). The paper is based upon the thesis [6], which builds upon the results of the paper...

Class groups of abelian fields, and the main conjecture

Cornelius Greither (1992)

Annales de l'institut Fourier

This first part of this paper gives a proof of the main conjecture of Iwasawa theory for abelian base fields, including the case p = 2 , by Kolyvagin’s method of Euler systems. On the way, one obtains a general result on local units modulo circular units. This is then used to deduce theorems on the order of χ -parts of p -class groups of abelian number fields: first for relative class groups of real fields (again including the case p = 2 ). As a consequence, a generalization of the Gras conjecture is stated...

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