zbMATH — the first resource for mathematics

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 \(K_{\infty}^{cycl}/K\).

11R32 Galois theory
11R23 Iwasawa theory
Full Text: DOI
[1] E. Artin J. Tate 1967
[2] Binz, A Subgroup Theorem of Profinite Groups, J. of Algebra 19 (1971)
[3] Brumer, On the Units of Algebraic Number Fields., Mathematika 14 pp 121– (1967) · Zbl 0171.01105
[4] A. Charifi p 1982
[5] G. Gras 2 1986
[6] Gras, Logarithme p-adique et groupes de Galois, J. für reine und angew. Math. 343 pp 64– (1983)
[7] G. Gras
[8] K. Haberland 1978
[9] H. Koch p 1970
[10] Miki, On the Leopoldt Conjecture on the p-adic Regulators, J. of Number Theory 26 pp 117– (1987) · Zbl 0621.12009
[11] Miki, Leopoldt’s Conjecture and Reiner’s Theorem, J. Math. Soc. Japan 86 (1) pp 47– (1984) · Zbl 0534.12005
[12] Neukirch, Freie Produkte pro-endlicher Gruppen und ihre Kohomologie, Arch. Math. 22 pp 337– (1971) · Zbl 0254.20023
[13] Neumann, On p-closed Number Fields and an Analogue of Riemann’s Existence Theorem in Algebraic Number Fields pp 625– (1977)
[14] Nguyen Quang Do, Sur la Zp-torsion de certains modules galoisiens, Ann. Inst. Fourier, Grenoble 36 pp 2– (1986) · Zbl 0576.12010
[15] Serre, Cohomologie galoisienne, Lecture Notes in Math. 5 (1965)
[16] Washington, G.T.M. n{\(\deg\)} 83, in: Introduction to Cyclotomic Fields (1982)
[17] Wingberg, Freie Produktzerlegungen von Galoisgruppen und lwasawa Invarianten für p-Erweiterungen von Q, J. für reine und angew. Math. 343 pp 111– (1983)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.