A relative class number formula for an imaginary abelian number field by means of Dedekind sum

2002 ◽  
Vol 109 (2) ◽  
pp. 223-227 ◽  
Author(s):  
Mikihito Hirabayashi
Keyword(s):  
Number Field ◽  
Class Number ◽  
Dedekind Sum ◽  
Relative Class ◽  
Class Number Formula ◽  
Relative Class Number ◽  
Number Formula ◽  
Abelian Number Field

scholarly journals Class number formulae in the form of a product of determinants in function fields

2005 ◽  
Vol 78 (2) ◽  
pp. 227-238 ◽  
Author(s):  
Jaehyun Ahn ◽  
Soyoung Choi ◽  
Hwanyup Jung
Keyword(s):  
Class Number ◽  
Prime Power ◽  
Function Fields ◽  
The Real ◽  
Relative Class ◽  
Intermediate Field ◽  
Class Number Formula ◽  
Relative Class Number ◽  
Number Formula

AbstractIn this paper, we generalize the Kučera's group-determinant formulae to obtain the real and relative class number formulae of any subfield of cyclotomic function fields with arbitrary conductor in the form of a product of determinants. From these formulae, we generalize the relative class number formula of Rosen and Bae-Kang and obtain analogous results of Tsumura and Hirabayashi for an intermediate field in the tower of cyclotomic function fields with prime power conductor.

IWASAWA TYPE FORMULA FOR COVERS OF A LINK IN A RATIONAL HOMOLOGY SPHERE

2008 ◽  
Vol 17 (10) ◽  
pp. 1199-1221 ◽  
Author(s):  
TERUHISA KADOKAMI ◽  
YASUSHI MIZUSAWA
Keyword(s):  
Number Field ◽  
Class Number ◽  
Homology Sphere ◽  
Rational Homology ◽  
Homology Groups ◽  
Cyclic Covers ◽  
Class Number Formula ◽  
Number Formula ◽  
Iwasawa Invariants ◽  
Rational Homology Sphere

Based on the analogy between links and primes, we present an analogue of the Iwasawa's class number formula in a Zp-extension for the p-homology groups of pn-fold cyclic covers of a link in a rational homology 3-sphere. We also describe the associated Iwasawa invariants precisely for some examples and discuss analogies with the number field case.

On the class number of the lpth cyclotomic number field

1991 ◽  
Vol 109 (2) ◽  
pp. 263-276
Author(s):  
Norikata Nakagoshi
Keyword(s):  
Number Field ◽  
Class Number ◽  
Prime Factor ◽  
Analytic Formula ◽  
Cyclotomic Number ◽  
Class Number Formula ◽  
Prime Factors ◽  
Number Formula ◽  
Genus Number ◽  
Analytic Class

The first factor of the class number of a cyclotomic number field can be obtainable by the analytic class number formula and there are some tables which show the decompositions of the first factors into primes. But, using just the analytic formula, we cannot tell what kinds of primes will appear as the factors of the class number of a given cyclotomic number field, except for those of the genus number, or the irregular primes. It is significant to find in advance the prime factors, particularly those prime to the degree of the field. For instance, in the table of the first factors we can pick out some pairs (l, p) of two odd primes l and p such that the class number of each lpth cyclotomic number field is divisible by l even if p 1 (mod l). If p ≡ (mod l) for l ≥ 5 or p ≡ 1 (mod 32) for l = 3, then it is easy from the outset to achieve our intention of finding the factor l using the genus number formula. Otherwise it seems to be difficult. We wish to make it clear algebraically why the class number has the prime factor l.

scholarly journals An application of the -adic analytic class number formula

2016 ◽  
Vol 19 (1) ◽  
pp. 217-228 ◽  
Author(s):  
Claus Fieker ◽  
Yinan Zhang
Keyword(s):  
Number Field ◽  
Class Number ◽  
Abelian Extension ◽  
Rational Field ◽  
Entire Class ◽  
Class Number Formula ◽  
Number Formula ◽  
Totally Real ◽  
Analytic Class

We propose an algorithm to verify the$p$-part of the class number for a number field$K$, provided$K$is totally real and an abelian extension of the rational field$\mathbb{Q}$, and$p$is any prime. On fields of degree 4 or higher, this algorithm has been shown heuristically to be faster than classical algorithms that compute the entire class number, with improvement increasing with larger field degrees.

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