On the rank of ideal class groups

1999 ◽  
pp. 377-383 ◽  
Author(s):  
Siman Wong
Keyword(s):  
Ideal Class ◽  
Class Groups ◽  
Ideal Class Groups

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.

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