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

## TL;DR

This paper derives a new class number formula for maximal real subfields of cyclotomic fields, introduces a novel type of cyclotomic units, and establishes a relation akin to Kummer's between class number and unit index.

## Contribution

It provides a new class number formula for real subfields of cyclotomic fields and constructs new cyclotomic units, extending Kummer's relation.

## Key findings

- Derived a formula for the special value of Euler-Dirichlet L-function at s=1
- Constructed a new type of cyclotomic units in $ ext{Q}( ext{μ}_{p^{n}})$
- Established a relation between class number and unit index similar to Kummer's

## Abstract

In this paper, we obtain a formula for the special value of Euler-Dirichlet $L$-function $L_E(s,\chi)$ at $s=1$. This leads to another class number formula of $\mathbb{Q}(\mu_{m})^{+}$, the maximal real subfield of $m$th cyclotomic field. From this formula, we construct a new type of cyclotomic units in $\mathbb{Q}(\mu_{p^{n}})$, which implies a similar Kummer's relation between the ideal class number of $\mathbb{Q}(\mu_{p^{n}})^{+}$ and the unit index.