## 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

### Citations:

Zbl 0509.12008; Zbl 0501.12015
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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 [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 [6] G. Gras : Sur les Z 2-extensions d’un corps quadratique imaginaire . Ann. Inst. Fourier 33 4 (1983) 1-18. · Zbl 0501.12016 [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 [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 [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
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