## 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

### Keywords:

pro-p-extension; p-rational; Leopoldt’s conjecture
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### References:

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