  For a number field K and a prime number p we denote by BP\_K the compositum
of the cyclic p-extensions of K embeddable in a cyclic p-extension of arbitrary
large degree. Then BP\_K is p-ramified (= unramified outside p) and is a finite
extension of the compositum K~ of the Z\_p-extensions of K.We study the
transfer map j\_(L/K) (as a capitulation map of ideal classes) for the
Bertrandias-Payan module bp\_K:=Gal(BP\_K/K~) in a p-extension L/K
(p\textgreater{}2, assuming the Leopoldt conjecture). In the cyclic case of
degree p, j\_(L/K) is injective except if L/K is kummerian, p-ramified, non
globally cyclotomic but locally cyclotomic at p (Theorem 3.1). We then intend
to characterize the condition \#bp\_K divides \#bp\_L^G (fixed points). So we
study bp\_L^G when j\_(L/K) is not injective and show that it depends on the
Galois group (over K~) of the maximal Abelian p-ramified pro-p-extension of
K.We give complete proofs in an elementary way using ideal approach of global
class field theory.
