Arithmetic characterization of algebraic number fields with small class numbers

1981 ◽  
Vol 176 (2) ◽  
pp. 247-253 ◽  
Author(s):  
Alfred Czogala
Keyword(s):  
Algebraic Number ◽  
Number Fields ◽  
Class Numbers ◽  
Small Class ◽  
Algebraic Number Fields

A note on class numbers of algebraic number fields

2001 ◽  
pp. 372-373
Author(s):  
Ichiro Satake ◽  
Genjiro Fujisaki ◽  
Kazuya Kato ◽  
Masato Kurihara ◽  
Shoichi Nakajima
Keyword(s):  
Algebraic Number ◽  
Number Fields ◽  
Class Numbers ◽  
Algebraic Number Fields

An arithmetic characterization of algebraic number fields with a given class group

1983 ◽  
Vol 94 (1) ◽  
pp. 23-28 ◽  
Author(s):  
David E. Rush
Keyword(s):  
Abelian Group ◽  
Algebraic Number ◽  
Class Group ◽  
Finite Abelian Group ◽  
Number Fields ◽  
Unique Factorization ◽  
Class Groups ◽  
Algebraic Number Fields ◽  
Ring Of Integers

Let R be the ring of integers of a number field K with class group G. It is classical that R is a unique factorization domain if and only if G is the trivial group; and the finite abelian group G is generally considered as a measure of the failure of unique factorization in R. The first arithmetic description of rings of integers with non-trivial class groups was given in 1960 by L. Carlitz (1). He proved that G is a group of order ≤ two if and only if any two factorizations of an element of R into irreducible elements have the same number of factors. In ((6), p. 469, problem 32) W. Narkiewicz asked for an arithmetic characterization of algebraic number fields K with class numbers ≠ 1, 2. This problem was solved for certain class groups with orders ≤ 9 in (2), and for the case that G is cyclic or a product of k copies of a group of prime order in (5). In this note we solve Narkiewicz's problem in general by giving arithmetical characterizations of a ring of integers whose class group G is any given finite abelian group.

A Note on the Class-Numbers of Algebraic Number Fields

1956 ◽  
Vol 78 (1) ◽  
pp. 51 ◽  
Author(s):  
N. C. Ankeny ◽  
R. Brauer ◽  
S. Chowla
Keyword(s):  
Algebraic Number ◽  
Number Fields ◽  
Class Numbers ◽  
Algebraic Number Fields
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