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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].}

MSC:

11R23 Iwasawa theory
11R18 Cyclotomic extensions
11R32 Galois theory
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References:

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