# The p-adic Kummer-Leopoldt constant - Normalized p-adic regulator

## TL;DR

This paper provides an elementary p-adic proof and improvements on the computation of the p-adic Kummer-Leopoldt constant, offering a class field theory interpretation and applications without relying on complex analytical methods.

## Contribution

It introduces a new elementary approach to the p-adic Kummer-Leopoldt constant, improves existing results, and defines a normalized p-adic regulator using class field theory techniques.

## Key findings

- Elementary proof of the p-adic Kummer-Leopoldt constant
- Improved bounds and calculations for the constant
- Application to generalizations of Kummer's lemma

## Abstract

The p-adic Kummer--Leopoldt constant kappa\_K of a number field K is (assuming the Leopoldt conjecture) the least integer c such that for all n \textgreater{}\textgreater{} 0, any global unit of K, which is locally a p^(n+c)th power at the p-places, is necessarily the p^nth power of a global unit of K. This constant has been computed by Assim \& Nguyen Quang Do using Iwasawa's techniques,after intricate studies and calculations by many authors. We give an elementary p-adic proof and an improvement of these results, then a class field theory interpretation of kappa\_K. We give some applications (including generalizations of Kummer's lemma on regular pth cyclotomic fields) and a natural definition of the normalized p-adic regulator for any K and any p$\ge$2.This is done without analytical computations, using only class field theoryand especially the properties of the so-called p-torsion group T\_K of Abelian p-ramification theory over K.