scholarly journals Indivisibility of the class number of a real abelian field of prime conductor

2021 ◽  
Vol 53 (0) ◽  
pp. 1-16
Author(s):  
Shoichi Fujima ◽  
Humio Ichimura
Keyword(s):  
Class Number ◽  
Abelian Field

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.

scholarly journals Note on class number parity of an abelian field of prime conductor

2018 ◽  
Vol 50 (0) ◽  
pp. 15-26
Author(s):  
Shoichi Fujima ◽  
Humio Ichimura
Keyword(s):  
Class Number ◽  
Abelian Field

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).

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