A complete parametriziation of cyclic field extensions of 2-power degree.
A determinant formula for the relative class number of an imaginary abelian number field
We give a new formula for the relative class number of an imaginary abelian number field by means of determinant with elements being integers of a cyclotomic field generated by the values of an odd Dirichlet character associated to . We prove it by a specialization of determinant formula of Hasse.
A finiteness theorem for imaginary abelian number fields.
A generalization of a result on integers in metacyclic extensions
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 generalization of a unit index of Greither
A generalization of Maillet and Demyanenko determinants
A note on normal and power bases
A note on Sinnott's index formula
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].
An algorithmic construction of cyclic p-extensions of fields, with characteristic different from p, not containing the pth roots of unity
An application of the p-adic class number formula.
Anneaux d’entiers stablement libres sur
Le groupe 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 ; cela se fait en liant un critère de Martinet avec une construction de Witt.
Annihilators for the class group of a cyclic field of prime power degree
Annihilators of minus class groups of imaginary abelian fields
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
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.
Arithmétique des corps abéliens du troisième degré
Average Frobenius distributions for elliptic curves over abelian extensions
Canonical factorisations in multiplicative Galois structure.
Capitulation and transfer kernels
If is a finite Galois extension of number fields with Galois group , then the kernel of the capitulation map of ideal class groups is isomorphic to the kernel of the transfer map where and is the Hilbert class field of . H. Suzuki proved that when is abelian, divides . We call a finite abelian group a transfer kernel for if for some group extension . After characterizing transfer kernels in terms of integral representations of , we show that is a transfer kernel for...
Circular units of real abelian fields with four ramified primes
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
This first part of this paper gives a proof of the main conjecture of Iwasawa theory for abelian base fields, including the case , 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 -parts of -class groups of abelian number fields: first for relative class groups of real fields (again including the case ). As a consequence, a generalization of the Gras conjecture is stated...