On Z p - embeddability of cyclic p - class fields ∗

It is investigated when a cyclic p-class field of an imaginary quadratic number field can be embedded in an infinite pro-cyclic p-extension. Résumé. On donne des conditions pour qu’un p-corps de classes cyclique d’un corps de nombres quadratique imaginaire soit plongeable dans une p-extension pro-cyclique infinie. Consider an imaginary quadratic number field K. Let p be an odd prime number, and denote by Zp the pro-cyclic pro-p-group lim ←−n(Z/p ). As shown by Iwasawa, any Zp-extension of K is unramified outside p. The lower steps of a such extension might well be unramified also at p. In this article the following question is investigated: if the p-class group of K is non-trivial and cyclic, is the p-Hilbert class field of K (or part of it) then embeddable in a Zp-extension of K? In doing so, we are led to study the torsion subgroup of the Galois group over K of the maximal abelian p-extension of K which is unramified outside p. First fix some notation: p : an odd prime number ζ : a primitive p-th root of unity ∆ : a square-free natural number K : the imaginary quadratic number field Q( √ −∆) O : the ring of integral elements in K K0 : the p-Hilbert class field of K Ke : the p-part of K’s ray class field with conductor p , e = 0 K∞ : the union ⋃∞ e=0Ke T : the torsion subgroup of Gal(K∞/K) K : the cyclotomic Zp-extension of K K : the anti-cyclotomic Zp-extension of K I : the group of fractional ideals of K prime to p ∗AMS subject classification: 11R32