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.

ON THE 2-ADIC IWASAWA LAMBDA INVARIANTS OF THE p-CYCLOTOMIC FIELDS AND THEIR QUADRATIC TWISTS

2014 ◽  
Vol 10 (02) ◽  
pp. 283-296
Author(s):  
HUMIO ICHIMURA ◽  
SHOICHI NAKAJIMA ◽  
HIROKI SUMIDA-TAKAHASHI
Keyword(s):  
Prime Number ◽  
Class Number ◽  
Cyclotomic Field ◽  
Cyclotomic Fields ◽  
Relative Class ◽  
Quadratic Twists ◽  
Large N ◽  
Relative Class Number

Let p be an odd prime number, Kn = Q(ζpn+1) the pn+1th cyclotomic field and [Formula: see text] the relative class number of Kn. Fixing an integer d ∈ Z with [Formula: see text], we denote by Ln the imaginary quadratic subextension of the imaginary (2, 2)-extension [Formula: see text] with Ln ≠ Kn. When d < 0, we have [Formula: see text]. Denote by [Formula: see text] and [Formula: see text] the minus parts of the 2-adic Iwasawa lambda invariants of Kn and Ln, respectively. By a theorem of Friedman, these invariants are stable for sufficiently large n. First, under the assumption that [Formula: see text] is odd for all n ≥ 1, we give a quite explicit version of this result. Second, we show that the assumption is satisfied for all p ≤ 599. Further, using these results, we compute the invariants [Formula: see text] and [Formula: see text] with d = -1, -3 for all p ≤ 599 and all n with the help of the computer.

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
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