Bases normales, unités et conjecture faible de Leopoldt. (Normal bases, units, and the weak Leopoldt conjecture). (French) Zbl 0732.11063

Let k be a number field, p an odd prime, \(R_ k={\mathcal O}_ k[p^{-1}]\) the ring of p-integers. Let \(Gal(R_ k,{\mathbb{Z}}_ p)=\lim_{\leftarrow} Gal(R_ k,C_ n)\), the limit of the groups of isomorphism classes of Galois extensions of \(R_ k\) with group \(C_ n={\mathbb{Z}}/p^ n{\mathbb{Z}}\), and let \(NB(R_ k,{\mathbb{Z}}_ p)=\lim_{\leftarrow}NB(R_ k,C_ n)\) where \(NB(R_ k,C_ n)\) is the subgroup of classes with normal basis. It is known that \(Gal(R_ k,{\mathbb{Z}}_ p)\) is a free \({\mathbb{Z}}_ p\)- module of rank \(1+r_ 2+\delta\), where \(r_ 2\) is the number of complex places of k and \(\delta\geq 0:\) Leopoldt’s conjecture is that \(\delta =0.\)
I. Kersten and J. Michaliček [J. Number Theory 32, 131- 150 (1989; Zbl 0709.11057)] showed that if k is a CM field, then \(NB(R_ k,{\mathbb{Z}}_ p)\) is a free \({\mathbb{Z}}_ p\)-module of rank \(1+r_ 2\). The authors extend this result to any number field k. Their approach is to split off the subgroup \(C_{\infty}\) of \(NB(R_ k,{\mathbb{Z}}_ p)\) generated by the cyclotomic \({\mathbb{Z}}_ p\)-extension. Let \(NB'(R_ k,{\mathbb{Z}}_ p)=NB(R_ k,{\mathbb{Z}}_ p)/C_{\infty}\), etc., then if \(K_ n=k[\mu_{p^ n}]\) and \(G_ n=Gal(K_ n/k)\), they obtain \(Gal'(R_ k,C_ n)=Gal'(R_{K_ n},C_ n)^{G_ n}\), so that \(Gal'(R_ k,C_ n)\) can be described for each n by descent from the exact sequence of Kummer theory. Thus \[ Gal'(R_ k,{\mathbb{Z}}_ p)=\lim_{\leftarrow} Gal'(R_{K_ n},C_ n)^{G_ n} \] and also for \(NB'\). This permits the application of Iwasawa theory to show that the rank of \(NB'(R_ k,{\mathbb{Z}}_ p)\) is \(1+r_ 2\) if and only if the weak Leopoldt conjecture holds for the cyclotomic \({\mathbb{Z}}_ p\)- extension of k. This latter result is known [c.f. V. Fleckinger C. R. Acad. Sci., Paris, Ser. I 302, 607-610 (1986; Zbl 0594.12006)].
{The same result has also been obtained by C. Greither [Habilitation, Univ. München, Jan. 1988].}


11R23 Iwasawa theory
11R18 Cyclotomic extensions
11R32 Galois theory
Full Text: DOI EuDML


[1] Brinkhuis, J.:”Embedding problems and Galois modules”, Thèse, Amsterdam (1981) · Zbl 0609.12008
[2] Chase, S. U., Harrison, D. K. & A. Rosenberg:Galois theory and Galois cohomology of commutative rings, Mem. Amer. Math. Soc., 52, 15–33 (1965) · Zbl 0143.05902
[3] Childs L. N.:Abelian Galois extensions of rings containing roots of unity, Illinois J. Math., 15, 273–279 (1971) · Zbl 0211.37102
[4] Childs, L. N.:The group of unramified Kummer extensions of prime degree, Proc. London Math. Soc., 35, 407–422 (1977) · Zbl 0374.13002
[5] Childs, L. N.:Cyclic Stickelberger cohomology and descent of Kummer extensions, Proc. Amer. Math. Soc., 90, 505–510 (1984) · Zbl 0538.12013
[6] Fleckinger, V.:Une interprétation de la conjecture de Leopoldt, C.R.Acad. Sc. Paris, t. 302, Série I n o 17,607–610 (1986) · Zbl 0594.12006
[7] Garfinkel, G. & Orzech, M.:Galois extensions as modules over the group ring, Canad. J. Math., 22, 505–510 (1970) · Zbl 0197.03401
[8] Greenberg, R.:A note on K 2 and the theory of Z p -extensions,Amer. J. Math.,100,n o 6,1235–1245 (1978) · Zbl 0408.12012
[9] Greither, C.:Galois extensions and normal bases, à paraître dans Trans. Amer. Math. Soc. · Zbl 0743.11060
[10] Grothendieck, A. & all:”S.G.A. 1”, Springer L.N.M. 224 (1971); ”S.G.A. 4”, Springer L.N.M. 270 (1972)
[11] Iwasawa, K.:On Z 1-extensions of algebraic number fields,Ann. Math.,98,246–326 (1973) · Zbl 0285.12008
[12] Kersten, I. & Michalicěk, J.:Z p-extensions of CM-fields, J. Number Theory, 32, n o 2,131–150 (1989) · Zbl 0709.11057
[13] Kersten, I. & Michalicěk, J.:On Vandiver’s conjecture and Z p-extensions of \(\mathbb{Q}(\zeta _{p^n } )\) ,J. Number Theory,32,n o 3,371–386 (1989) · Zbl 0709.11058
[14] Kuz’min, L. V.:The tate module for algebraic number fields, Math. U.S.S.R. Izv., 6, n o 2,267–321 (1972)
[15] Nguyen Quang Do, T.:Sur la cohomologie de certains modules galoisiens p-ramifiés, dans J.-M. de Koninck & C. Levesque (éd.) ”Théorie des Nombres”, Actes de la conférence de Laval 91987), W. de Gruyter, 740–754 (1989). · Zbl 0697.12009
[16] Nguyen Quang Do, T.:Sur la torsion de certains modules galoisiens II, Sém. Théorie des Nombres de Paris 1986–87, Birkhäuser, 271–297 (1989) · Zbl 0576.12010
[17] Serre, J.-P.: ”Corps locaux”,Paris, Hermann (1968)
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.