An analogue of Kummer's relation between the ideal class number and the unit index of cyclotomic fields

2019/02/02 by Hu, Su, Kim, Min-Soo, Li, Yan
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1902.00718

Abstract

In this paper, we obtain a formula for the special value of Euler-Dirichlet L-function L_E(s,χ) at s=1. This leads to another class number formula of ℚ(μ_m)⁺, the maximal real subfield of mth cyclotomic field. From this formula, we construct a new type of cyclotomic units in ℚ(μ_pⁿ), which implies a similar Kummer's relation between the ideal class number of ℚ(μ_pⁿ)⁺ and the unit index.

Related

• The first factor of the class number of the p-th cyclotomic field
• On the Applications of Cyclotomic Fields in Introductory Number Theory
• Ideal class groups of cyclotomic number fields I
• Computation of the Kummer ratio of the class number for prime cyclotomic fields
• Relative class numbers and Euler-Kronecker constants of maximal real cyclotomic subfields