On minus quotients of ideal class groups of cyclotomic fields

2020 ◽  
Vol 16 (09) ◽  
pp. 2013-2026
Author(s):  
Satoshi Fujii
Keyword(s):  
Abelian Group ◽  
Class Group ◽  
Finite Abelian Group ◽  
Cyclotomic Field ◽  
Cyclotomic Fields ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Ideal Class Groups ◽  
The Ideal

Let [Formula: see text] be the minus quotient of the ideal class group of the [Formula: see text]th cyclotomic field. In this paper, first, we show that each finite abelian group appears as a subgroup of [Formula: see text] for some [Formula: see text]. Second, we show that, for all pairs of integers [Formula: see text] and [Formula: see text] with [Formula: see text], the kernel of the lifting map [Formula: see text] is contained in the [Formula: see text]-torsion [Formula: see text] of [Formula: see text]. Such an evaluation of the exponent is an individuality of cyclotomic fields.

scholarly journals A Note on Ideal Class Groups

1966 ◽  
Vol 27 (1) ◽  
pp. 239-247 ◽  
Author(s):  
Kenkichi Iwasawa
Keyword(s):  
Class Group ◽  
Cyclotomic Field ◽  
Theoretical Method ◽  
Number Fields ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Algebraic Number Fields ◽  
Ideal Class Groups ◽  
The Ideal

In the first part of the present paper, we shall make some simple observations on the ideal class groups of algebraic number fields, following the group-theoretical method of Tschebotarew. The applications on cyclotomic fields (Theorems 5, 6) may be of some interest. In the last section, we shall give a proof to a theorem of Kummer on the ideal class group of a cyclotomic field.

scholarly journals On nilpotent factors of congruent ideal class groups of Galois extensions

1976 ◽  
Vol 62 ◽  
pp. 13-28 ◽  
Author(s):  
Yoshiomi Furuta
Keyword(s):  
Class Group ◽  
Algebraic Number Field ◽  
Augmentation Ideal ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Finite Degree ◽  
Ring Of Integers ◽  
Ideal Class Groups ◽  
The Ideal

Let K be a Galois extension of an algebraic number field k of finite degree with Galois group g. Then g acts on a congruent ideal class group of K as a group of automorphisms, when the class field M over K corresponding to is normal over K. Let Ig be the augmentation ideal of the group ring Zg over the ring of integers Z, namely Ig be the ideal of Zg generated by σ − 1, σ running over all elements of g. Then is the group of all elements aσ-1 where a and σ belong to and g respectively.

On the orders of ideal classes in prime cyclotomic fields

1990 ◽  
Vol 108 (2) ◽  
pp. 197-201 ◽  
Author(s):  
Francisco Thaine
Keyword(s):  
Sylow Subgroup ◽  
Class Group ◽  
Cyclotomic Field ◽  
Cyclotomic Fields ◽  
Ideal Class ◽  
Ideal Class Group ◽  
The Ideal

In this article we exhibit a method complementary to the method presented in [4], that allows us, at least in some important cases, to obtain exact expressions for the orders of ideal classes of cyclotomic fields in terms of properties of the units of the field. We consider only the particular case in which the classes belong to the p-Sylow subgroup (A)p of the ideal class group of a real p-cyclotomic field, but it appears that the results can be generalized.

scholarly journals The 2-Ideal Class Groups of ℚ(ζl)

2001 ◽  
Vol 162 ◽  
pp. 1-18 ◽  
Author(s):  
Pietro Cornacchia
Keyword(s):  
Class Group ◽  
Ideal Class ◽  
Galois Module ◽  
Ideal Class Group ◽  
Class Groups ◽  
Ideal Class Groups ◽  
The Ideal

For prime l we study the structure of the 2-part of the ideal class group Cl of ℚ(ζl). We prove that Cl ⊗ ℤ2) is a cyclic Galois module for all l < 10000 with one exception and compute the explicit structure in several cases.

scholarly journals On the Galois module structure of ideal class groups

2001 ◽  
Vol 164 ◽  
pp. 133-146 ◽  
Author(s):  
Toru Komatsu ◽  
Shin Nakano
Keyword(s):  
Class Group ◽  
Class Numbers ◽  
Module Structure ◽  
Ideal Class ◽  
Galois Module ◽  
Ideal Class Group ◽  
Class Groups ◽  
Prime Degree ◽  
Ideal Class Groups ◽  
The Ideal

Let K/k be a Galois extension of a number field of degree n and p a prime number which does not divide n. The study of the p-rank of the ideal class group of K by using those of intermediate fields of K/k has been made by Iwasawa, Masley et al., attaining the results obtained under respective constraining assumptions. In the present paper we shall show that we can remove these assumptions, and give more general results under a unified viewpoint. Finally, we shall add a remark on the class numbers of cyclic extensions of prime degree of Q.

The 2-Rank of the Class Group of Imaginary Bicyclic Biquadratic Fields

1997 ◽  
Vol 49 (2) ◽  
pp. 283-300 ◽  
Author(s):  
Thomas M. McCall ◽  
Charles J. Parry ◽  
Ramona R. Ranalli
Keyword(s):  
Sylow Subgroup ◽  
Class Group ◽  
Class Numbers ◽  
Ideal Class ◽  
Small Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Ideal Class Groups ◽  
Biquadratic Fields ◽  
The Ideal

AbstractA formula is obtained for the rank of the 2-Sylow subgroup of the ideal class group of imaginary bicyclic biquadratic fields. This formula involves the number of primes that ramify in the field, the ranks of the 2-Sylow subgroups of the ideal class groups of the quadratic subfields and the rank of a Z2-matrix determined by Legendre symbols involving pairs of ramified primes. As applications, all subfields with both 2- class and class group Z2×Z2 are determined. The final results assume the completeness of D. A. Buell’s list of imaginary fields with small class numbers.

scholarly journals Tournaments and Ideal Class Groups

1995 ◽  
Vol 38 (3) ◽  
pp. 330-333
Author(s):  
Robert J. Kingan
Keyword(s):  
Class Group ◽  
Number Fields ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Quadratic Number ◽  
Class Groups ◽  
Ideal Class Groups ◽  
Quadratic Number Fields ◽  
The Ideal

AbstractResults are given for a class of square {0,1}-matrices which provide information about the 4-rank of the ideal class group of certain quadratic number fields.

Circulant Graphs and 4-Ranks of Ideal Class Groups

1994 ◽  
Vol 46 (1) ◽  
pp. 169-183 ◽  
Author(s):  
Jurgen Hurrelbrink
Keyword(s):  
Quadratic Forms ◽  
Class Group ◽  
Number Fields ◽  
Regular Graphs ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Quadratic Number ◽  
Ideal Class Groups ◽  
Yield Information ◽  
The Ideal

AbstractThis is about results on certain regular graphs that yield information about the structure of the ideal class group of quadratic number fields associated with these graphs. Some of the results can be formulated in terms of the quadratic forms x2 + 27y2, x2 + 32y2, x2 + 64y2.

scholarly journals On the central ideal class group of cyclotomic fields

1979 ◽  
Vol 75 ◽  
pp. 133-143 ◽  
Author(s):  
Susumu Shirai
Keyword(s):  
Rational Number ◽  
Class Group ◽  
Abelian Extension ◽  
Factor Group ◽  
Augmentation Ideal ◽  
Cyclotomic Fields ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Abelian Field ◽  
The Ideal

Let Q be the rational number field, K/Q be a finite Galois extension with the Galois group G, and let CK be the ideal class group of K in the wider sense. We consider CK as a G-module. Denote by I the augmentation ideal of the group ring of G over the ring of rational integers. Then CK/I(CK) is called the central ideal class group of K, which is the maximal factor group of CK on which G acts trivially. A. Fröhlich [3, 41 rationally determined the central ideal class group of a complete Abelian field over Q whose degree is some power of a prime. The proof is based on Theorems 3 and 4 of Fröhlich [2]. D. Garbanati [6] recently gave an algorithm which will produce the l-invariants of the central ideal class group of an Abelian extension over Q for each prime l dividing its order.

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