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Plongements Kummeriens dans les \({\mathbb{Z}}_ p\)-extensions. (Kummer embeddings in \({\mathbb{Z}}_ p\)-extensions). (English) Zbl 0584.12004

Let k be a number field containing the group of \(p^ e\)-th roots of unity (where p is a fixed prime and \(e\geq 1)\). Let \(\tilde k\) denote the composite of all \({\mathbb{Z}}_ p\)-extensions of k; then the maximal subextension \(\tilde N\) of \(\tilde k\) of exponent \(p^ e\) is of the form \(k(^{p^ e}\sqrt{\tilde R})\). Let S be the set of primes of k lying above p, and I the group of ideals of k of which a p-power is generated by an element \(\equiv 1\) mod\(\prod_{{\mathfrak p}\in S}{\mathfrak p}.\)
In a preceding paper [J. Reine Angew. Math. 343, 64-80 (1983; Zbl 0501.12015)], the author defined a p-adic logarithm function Log: \(I\to (\prod_{{\mathfrak p}\in S}k_{{\mathfrak p}})/\Lambda\), where \(\Lambda\) is a \({\mathbb{Q}}_ p\)-algebra generated by the image of units \(\equiv 1\) mod\(\prod_{{\mathfrak p}\in S}{\mathfrak p} \); and, for a p-ramified finite p- abelian extension M of k, gave an expression of the Artin group of \(M\cap \tilde k\) using Log.
In the present paper, using this result, the author gives an algorithm to calculate a set of generators of \(\tilde R\) from the p-class group and units of k which consists of the determination of the Artin group \(\tilde A\) of \(\tilde N\) by calculating Log, and the computation of \(\tilde R\) from \(\tilde A\) using the p-power residue symbol through Kummer theory. Especially, in the case that \(p^ e=2\) and k is imaginary quadratic, he computes explicitly several steps of this, and obtains an effective and systematic method to calculate \(\tilde R\) from a set of generators of the 2-class group of k. Several examples are also given.
Reviewer: T.Takeuchi

MSC:

11R18 Cyclotomic extensions
11S15 Ramification and extension theory
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
11R23 Iwasawa theory
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References:

[1] F. Bertrandias et J.-J. Payan : \Gamma -extensions et invariants cyclotomiques . Ann. Sci. Ec. Norm. Sup. 5 (1972) 517-543. · Zbl 0246.12005 · doi:10.24033/asens.1236
[2] J.E. Carroll : On the 2 - primary part of K2O and on Z2- extensions for imaginary quadratic fields , Ph. D. Harvard (1973).
[3] J.E. Carroll : On determining the quadratic subfields of Z 2-extensions of complex quadratic fields . Comp. Math. 30 3 (1975) 259-271. · Zbl 0314.12005
[4] J.E. Carroll and H. Kisilevsky : Initial layers of Z l-extensions of complex quadratic fields . Comp. Math. 32 2 (1976) 157-168. · Zbl 0357.12003
[5] G. Gras : Logarithme p-adique et groupes de Galois . Journal de Crelle 343 (1983) 64-80. · Zbl 0501.12015 · doi:10.1515/crll.1983.343.64
[6] G. Gras : Sur les Z 2-extensions d’un corps quadratique imaginaire . Ann. Inst. Fourier 33 4 (1983) 1-18. · Zbl 0501.12016 · doi:10.5802/aif.939
[7] R. Greenberg : A note on K2 and the theory of Z p-extensions . Amer. J. Math. 100 6 (1978) 1235-1245. · Zbl 0408.12012 · doi:10.2307/2373971
[8] K. Kramer and A. Candiotti : On K2 and Z l-extensions of number fields . Amer. J. Math. 100 1 (1978) 177-196. · Zbl 0388.12004 · doi:10.2307/2373879
[9] J.-P. Serre : Corps locaux . Hermann (1962). · Zbl 0137.02601
[10] J. Tate : Relations between K2 and Galois cohomology . Invent. Math. 36 (1976) 257-274. · Zbl 0359.12011 · doi:10.1007/BF01390012
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