scholarly journals A characterization of algebraic number fields with class number two.

1960 ◽  
Vol 11 (3) ◽  
pp. 391-391 ◽  
Author(s):  
L. Carlitz
Keyword(s):  
Algebraic Number ◽  
Class Number ◽  
Number Fields ◽  
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.

scholarly journals A note on the characterization of CM-fields

1978 ◽  
Vol 26 (1) ◽  
pp. 26-30 ◽  
Author(s):  
P. E. Blanksby ◽  
J. H. Loxton
Keyword(s):  
Unit Circle ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Subject Classification ◽  
Algebraic Number Fields

AbstractThis note deals with some properties of algebraic number fields generated by numbers having all their conjugates on a circle. In particular, it is shown that an algebraic number field is a CM-field if and only if it is generated over the rationals by an element (not equal to ±1) whose conjugate all lie on the unit circle.Subject classification (Amer. Math. Soc. (MOS) 1970): 12 A 15, 12 A 40, 14 K 22.

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