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Genre des corps surcirculaires. (Genus of supercircular fields). (French) Zbl 0614.12006
Publ. Math. Fac. Sci. Besançon, Théor. Nombres 1984/85-1985/86, No. 2, Exp. No. 3, 39 p. (1986).
This article is devoted to Iwasawa theory and more precisely to genus theory for cyclotomic $${\mathbb{Z}}_ p$$-fields in comparison with genus theory for function fields. The point is the discussion of analogues of the Riemann-Hurwitz formula and the Chevalley-Weil one for Galois p- extensions of $${\mathbb{Z}}_ p$$-fields. Such analogues were found by L. V. Kuz’min [Math. USSR, Izv. 14, 441-498 (1980); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 43, 483-546 (1979; Zbl 0434.12006)], Y. Kida [J. Number Theory 12, 519-528 (1980; Zbl 0455.12007)] and K. Iwasawa [Tôhoku Math. J., II. Ser. 33, 263-288 (1981; Zbl 0468.12004)]. Previous improvements of these results were given by K. Wingberg [Compos. Math. 55, 333-381 (1985; Zbl 0608.12012)], J. G. D’Mello and M. L. Madan [Manuscr. Math. 41, 75-107 (1983; Zbl 0516.12012)], and R. Gold and M. L. Madan [Commun. Algebra 13, 1559-1578 (1985; Zbl 0568.12003)].
The main originality of this work is to reduce all these formulae to one of them which can be proved in two different ways by using only the classical genus theory for number fields. Proofs rest on some of the main results of the author’s thesis [Publ. Math. Fac. Sci. Besançon, Théor. Nombres 1984/85-1985/86, No.1 (1986; Zbl 0601.12002)].

MSC:
 11R18 Cyclotomic extensions 11R58 Arithmetic theory of algebraic function fields 11R37 Class field theory