Title: The p-adic Kummer-Leopoldt constant - Normalized p-adic regulator,
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.