scholarly journals Formulae for the relative class number of an imaginary abelian field in the form of a determinant

2001 ◽  
Vol 163 ◽  
pp. 167-191 ◽  
Author(s):  
Radan Kučera
Keyword(s):  
Class Number ◽  
Relative Class ◽  
Abelian Field ◽  
Relative Class Number ◽  
Stickelberger Ideal ◽  
Cyclotomic Units

There is in the literature a lot of determinant formulae involving the relative class number of an imaginary abelian field. Usually such a formula contains a factor which is equal to zero for many fields and so it gives no information about the class number of these fields. The aim of this paper is to show a way of obtaining most of these formulae in a unique fashion, namely by means of the Stickelberger ideal. Moreover some new and non-vanishing formulae are derived by a modification of Ramachandra’s construction of independent cyclotomic units.

An extended Stickelberger ideal of the compositum of a bicyclic field and an imaginary quadratic field

2013 ◽  
Vol 63 (1) ◽  
Author(s):  
Veronika Trnková
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Relative Class ◽  
Relative Class Number ◽  
Stickelberger Ideal ◽  
Imaginary Field ◽  
Divisibility Properties

AbstractWe consider certain extension of the Stickelberger ideal of the compositum of a bicyclic field and a quadratic imaginary field, obtained by adding new annihilators to the Stickelberger ideal. We compute the index of this extension, from which we get some divisibility properties for the relative class number of the compositum.

scholarly journals A criterion for the parity of the class number of an Abelian field with prime power conductor

1997 ◽  
Vol 145 ◽  
pp. 163-177 ◽  
Author(s):  
Ken-Ichi Yoshino
Keyword(s):  
Class Number ◽  
Prime Power ◽  
Cyclotomic Field ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Relative Class ◽  
Abelian Field ◽  
Relative Class Number ◽  
Necessary And Sufficient

Let f be a positive integer such that f ≢ 2 (mod 4). Let h0 be the class number of the maximal real subfield of fth cyclotomic field Q(ζf)- It is interesting to determine when h0 is even. Kummer [11] investigated this problem when f is a prime and showed that if h0 is even, then the relative class number h of the cyclotomic field is even (Satz III). Moreover he gave another necessary condition for h0 to be even (Satz IV). In [7] Hasse gave a necessary and sufficient condition for h to be even (Satz 45).

Another look at real quadratic fields of relative class number 1

2014 ◽  
Vol 163 (4) ◽  
pp. 371-377 ◽  
Author(s):  
Debopam Chakraborty ◽  
Anupam Saikia
Keyword(s):  
Class Number ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Relative Class ◽  
Relative Class Number ◽  
Real Quadratic

scholarly journals On Dirichletʼs conjecture on relative class number one

2012 ◽  
Vol 132 (7) ◽  
pp. 1398-1403 ◽  
Author(s):  
Amanda Furness ◽  
Adam E. Parker
Keyword(s):  
Class Number ◽  
Relative Class ◽  
Relative Class Number

A Generalization of Jakubec's Formula

2015 ◽  
Vol 65 (1) ◽  
Author(s):  
Mikihito Hirabayashi
Keyword(s):  
Class Number ◽  
Cyclotomic Field ◽  
Not Given ◽  
Relative Class ◽  
Relative Class Number

AbstractIn 2009 Jakubec gave two determinantal formulas for the relative class number of the pth cyclotomic field, p an odd prime. We generalize one of the formulas to an arbitrary cyclotomic field and also determine the sign of the formula, which he had not given.

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