Complex Analytic Class Number Formulas

1978 ◽  
pp. 69-93
Author(s):  
Serge Lang
Keyword(s):  
Class Number ◽  
Analytic Class

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.

The Analytic Class Number Formula

2008 ◽  
pp. 185-207
Author(s):  
Michael J. Jacobson ◽  
Hugh C. Williams
Keyword(s):  
Class Number ◽  
Class Number Formula ◽  
Number Formula ◽  
Analytic Class

scholarly journals The analytic class number formula for 1‐dimensional affine schemes

2020 ◽  
Vol 52 (5) ◽  
pp. 793-806
Author(s):  
Bruce W. Jordan ◽  
Bjorn Poonen
Keyword(s):  
Class Number ◽  
Class Number Formula ◽  
Number Formula ◽  
Analytic Class

Analytic class number formulas in function fields

1981 ◽  
Vol 65 (1) ◽  
pp. 49-69 ◽  
Author(s):  
David R. Hayes
Keyword(s):  
Class Number ◽  
Function Fields ◽  
Analytic Class

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.

Note on class number parity of an abelian field of prime conductor, II

2019 ◽  
Vol 42 (1) ◽  
pp. 99-110 ◽  
Author(s):  
Humio Ichimura
Keyword(s):  
Class Number ◽  
Abelian Field
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