scholarly journals A note on the ideal class group of the cyclotomic Zp-extension of a totally real number field

2002 ◽  
Vol 105 (1) ◽  
pp. 29-34 ◽  
Author(s):  
Humio Ichimura
Keyword(s):  
Real Number ◽  
Number Field ◽  
Class Group ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Totally Real ◽  
Real Number Field ◽  
The Ideal

On a Theorem of Kawamoto on Normal Bases of Rings of Integers, II

2005 ◽  
Vol 48 (4) ◽  
pp. 576-579 ◽  
Author(s):  
Humio Ichimura
Keyword(s):  
Prime Number ◽  
Number Field ◽  
Class Group ◽  
Natural Homomorphism ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Integral Basis ◽  
Rings Of Integers ◽  
Integer Ring ◽  
The Ideal

AbstractLet m = pe be a power of a prime number p. We say that a number field F satisfies the property when for any a ∈ F×, the cyclic extension F(ζm, a1/m)/F(ζm) has a normal p-integral basis. We prove that F satisfies if and only if the natural homomorphism is trivial. Here K = F(ζm), and denotes the ideal class group of F with respect to the p-integer ring of F.

scholarly journals Subexponential class group and unit group computation in large degree number fields

2014 ◽  
Vol 17 (A) ◽  
pp. 385-403 ◽  
Author(s):  
Jean-François Biasse ◽  
Claus Fieker
Keyword(s):  
Number Field ◽  
Riemann Hypothesis ◽  
Class Group ◽  
Large Degree ◽  
Unit Group ◽  
Number Fields ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Subexponential Class ◽  
The Ideal

AbstractWe describe how to compute the ideal class group and the unit group of an order in a number field in subexponential time. Our method relies on the generalized Riemann hypothesis and other usual heuristics concerning the smoothness of ideals. It applies to arbitrary classes of number fields, including those for which the degree goes to infinity.

scholarly journals On Modified Circular Units and Annihilation of real Classes

2005 ◽  
Vol 177 ◽  
pp. 77-115 ◽  
Author(s):  
Jean-Robert Belliard ◽  
Thống Nguyễn-Quang-Ðỗ
Keyword(s):  
Real Number ◽  
Prime Number ◽  
Number Field ◽  
Class Group ◽  
Euler Systems ◽  
Totally Real ◽  
Circular Units ◽  
Functorial Approach ◽  
Real Number Field

For an abelian totally real number field F and an odd prime number p which splits totally in F, we present a functorial approach to special “p-units” previously built by D. Solomon using “wild” Euler systems. This allows us to prove a conjecture of Solomon on the annihilation of the p-class group of F (in the particular context here), as well as related annihilation results and index formulae.

scholarly journals Computation of L(0, χ) and of relative class numbers of CM-fields

2001 ◽  
Vol 161 ◽  
pp. 171-191 ◽  
Author(s):  
Stéphane Louboutin
Keyword(s):  
Real Number ◽  
Number Field ◽  
Class Group ◽  
Number Fields ◽  
Class Numbers ◽  
Finite Part ◽  
Relative Class ◽  
Relative Class Number ◽  
Totally Real ◽  
Real Number Field

Let χ be a nontrivial Hecke character on a (strict) ray class group of a totally real number field L of discriminant dL. Then, L(0, χ) is an algebraic number of some cyclotomic number field. We develop an efficient technique for computing the exact values at s = 0 of such abelian Hecke L-functions over totally real number fields L. Let fχ denote the norm of the finite part of the conductor of χ. Then, roughly speaking, we can compute L(0, χ) in O((dLfx)0.5+∊) elementary operations. We then explain how the computation of relative class numbers of CM-fields boils down to the computation of exact values at s = 0 of such abelian Hecke L-functions over totally real number fields L. Finally, we give examples of relative class number computations for CM-fields of large degrees based on computations of L(0, χ) over totally real number fields of degree 2 and 6.

CONOYAUX DE CAPITULATION ET MODULES D'IWASAWA

2011 ◽  
Vol 07 (06) ◽  
pp. 1503-1517
Author(s):  
FRÉDÉRIC PITOUN
Keyword(s):  
Real Number ◽  
Galois Group ◽  
Number Field ◽  
Class Group ◽  
Primary Roots ◽  
Totally Real ◽  
Cyclotomic Extension ◽  
Real Number Field

Soit F un corps de nombres totalement réel et p un premier impair, on note K0 = F(ζp). Pour n ∈ ℕ, Kn désigne le n-ième étage de la ℤp-extension cyclotomique K∞/K0, An est la p-partie du groupe des classes de Kn, [Formula: see text] et N∞ est l'extension de K∞ obtenue en extrayant des racines p-primaires d'unités. Le but de cet article est de montrer que le dual de Pontryagin de la partie plus des conoyaux de capitulation [Formula: see text], sur laquelle l'action de Γ a été tordue une fois par le caractère cyclotomique et la partie moins de la ℤp-torsion du groupe de Galois Gal (N∞ ∩ L∞/K∞) sont isomorphes. Let F be a totally real number field and p an odd prime, we note K0 = F(ζp). For an integer n, Kn is the nth floor of the ℤp-cyclotomic extension K∞/K0, An is the p-part of the class group of Kn, [Formula: see text] and N∞ is the extension of K∞ generated by p-primary roots of units. In this article, we prove that the plus part of the capitulation's cokernel [Formula: see text], on which Γ-action was twisted on time by the cyclotomic character, and the minus part of the ℤp-torsion of the Galois group Gal (N∞ ∩ L∞/K∞) is isomorphic.

Minimal generators of the ideal class group

2021 ◽  
Vol 222 ◽  
pp. 157-167
Author(s):  
Henry H. Kim
Keyword(s):  
Class Group ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Minimal Generators ◽  
The Ideal

scholarly journals On Certain Multiple Series with Functional Equation in a Totally Real Number Field II

1997 ◽  
Vol 20 (2) ◽  
pp. 299-313
Author(s):  
Takayoshi MITSUI
Keyword(s):  
Functional Equation ◽  
Real Number ◽  
Number Field ◽  
Multiple Series ◽  
Totally Real ◽  
Real Number Field
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