scholarly journals On an index formula for the relative class number of a cyclotomic number field

1990 ◽  
Vol 36 (3) ◽  
pp. 332-338
Author(s):  
Akira Endô
Keyword(s):  
Number Field ◽  
Class Number ◽  
Index Formula ◽  
Cyclotomic Number ◽  
Relative Class ◽  
Relative Class Number

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

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.

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