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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 ap^(n+c)th power at the p-places, is necessarily the p^nth power of a globalunit of K. This constant has been computed by Assim \& Nguyen Quang Do usingIwasawa's techniques,after intricate studies and calculations by many authors.We give an elementary p-adic proof and an improvement of these results, then aclass 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 anyp$\ge$2.This is done without analytical computations, using only class fieldtheoryand especially the properties of the so-called p-torsion group T\_K ofAbelian p-ramification theory over K.

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