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Sur les p-extensions des corps p-rationnels. (On p-extensions of p- rational fields). (French) Zbl 0723.11054
Let K be a number field, p a prime number, \(S_ p\) the set of places of K dividing p, S a finite set of places of K containing \(S_ p\), and \(c=c_ K\) the number of complex places of K. \(K_ S\) (resp. \(K_ S^{ab})\) denotes the maximal (resp. maximal abelian) S-ramified pro-p- extension of K and \(G_ S=G_ S(K)\) (resp. \(X_ S=X_ S(K))\) is the Galois group \(Gal(K_ S/K)\) (resp. \(Gal(K_ S^{ab}/K))\). Throughout we assume that \(p\neq 2\) or K is totally imaginary. K is called p-rational if K satisfies Leopoldt’s conjecture at p and the \({\mathbb{Z}}_ p\)-torsion subgroup of \(X_ S(K)\) is trivial. Examples are \({\mathbb{Q}}\) for all p, \({\mathbb{Q}}(\mu_{p^ n})\) for a regular prime p, and imaginary quadratic K for almost all p. Other characterizations (in terms of \(G_{S_ p},X_{S_ p}\) and c) for p-rationality can be given. If K/L is a Galois extension of number fields and K is p-rational, then so is L.
Now, let \(K_ S(1)\) denote the maximal abelian S-ramified extension of K such that \(Gal(K_ S(1)/K)\) has exponent p. S is called primitive for (K,p) if the system \(\{\sigma_{{\mathcal Q}}(K_{S_ p}(1)/K)| {\mathcal Q}\) a finite prime in \(S\setminus S_ p\) with norm N\({\mathcal Q}\equiv 1 mod p\}\) is free of rank \(| S\setminus S_ p|\) in the \({\mathbb{F}}_ p\)-vector space \(Gal(K_{S_ p}(1)/K)\), where \(\sigma_{{\mathcal Q}}(K_{S_ p}(1)/K)\) is the Frobenius of \({\mathcal Q}\) for the extension \(K_{S_ p}(1)/K\). Then
(i): K is p-rational and S is primitive for (K,p) iff
(ii): The pro-p-group \(G_ S\) is isomorphic to the free pro-p-product \(*_{{\mathcal Q}}Gal(\hat K_{{\mathcal Q}}K_{{\mathcal Q}})*F\), where F is the free group on \(d=\max \{0,1+c-| S\setminus S_ p| \}\) generators.
As a corollary one obtains: (i) (or (ii)) is equivalent to
(iii): Every S-ramified p-extension L of K satisfies Leopoldt’s conjecture.
For p-rational K containing \(\mu_ p\) one can calculate primitive S by means of tame symbols. Another result is the following theorem:
Let L/K be a p-extension of number fields. Then L is p-rational iff K is p-rational and the set S of places of K that are ramified in L, or divide p, is primitive for (K,p). Moreover, under these conditions, the extension of S to L is primitive for (L,p).
As a corollary one gets that, for a p-extension L/K of totally real number fields, L is p-rational iff K is p-rational and L/K is \(S_ p\cup {\mathcal Q}\)-ramified, where \({\mathcal Q}\not\in S_ p\) is inert in the cyclotomic \({\mathbb{Z}}_ p\)-extension \(K_{\infty}^{cycl}/K\).

MSC:
11R32 Galois theory
11R23 Iwasawa theory
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