The p-adic Kummer-Leopoldt constant -- Normalized p-adic regulator
Résumé
The p-adic Kummer--Leopoldt constant kappa(K)
of a number field K is (under the Leopoldt conjecture for p in K) the
least integer c such that for all n large enough, any global unit
epsilon of K, which is locally (at the p-places) a p^{n+c}th power, is
necessarily a p^{n}th power of a global unit of K. This constant has been
computed by J. Assim and T. Nguyen Quang Do using Iwasawa's technics.
In this short Note, we give an elementary p-adic proof of their result and
the class field theory interpretation of kappa(K) by means of
Abelian p-ramification theory over K. Then, from this, we give a natural
definition of the normalized p-adic regulator of K, for any K and any p≥2.
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