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Dualité dans les corps surcirculaires. (Duality in supercyclotomic fields). (French) Zbl 0679.12007
Sémin. Théor. Nombres, Paris 1986-87, Prog. Math. 75, 183-220 (1988).
[For the entire collection see Zbl 0653.00005.]
This article discusses some generalizations of the so-called Spiegelungssatz of H. W. Leopoldt for number fields and cyclotomic $${\mathbb{Z}}_{\ell}$$-extensions of number fields, including duality results of L. V. Kuz’min [Math. USSR, Izv. 14, 441-498 (1980; Zbl 0448.12007)], T. Nguyen Quang Do [Ann. Inst. Fourier 36, No.2, 27-46 (1986; Zbl 0576.12010)], and the author [Publ. Math. Fac. Sci. Besançon, Théor. Nombres, 1984/85-1985/86, No.1 (1986; Zbl 0601.12002)].
The main tool is the infinitesimal description of Galois groups and Kummer radicals introduced in a previous article [Ann. Inst. Fourier 34, No.2, 1-27 (1984; Zbl 0522.12014)], which provides a canonical isomorphism between some well-known modules, and gives rise to an analogue for cyclotomic $${\mathbb{Z}}_{\ell}$$-fields of the classical Weil pairing in function fields.
Reviewer: J.-F.Jaulent

##### MSC:
 11R18 Cyclotomic extensions