Ramifications minimales
Résumé
The author applies the so-called reflexion principle given by the Kummer duality in order to give a very simple description of cyclic p-extensions of number fields with minimal ramification by way of a governing field. More precisely let p be a prime number and S an arbitrary finite set of non-complex places of a given number field K. The so-called governing field for (K, S, p) is the p-extension F(S):=K[ζ_p , p√Y^S ] generated by p-th roots of unity ζ_p and the pth radical of Y^S := {yK^p ∈ K/K^p | ∀p outside S, v_p (y) ≡ 0 mod p} (where for p = 2 the condition implies y ≥p 0 at the real places p outside S ). This definition extends the previous one given for empty S in an article by the author and A. Munnier [in Théorie des nombres, Années 1996/97–1997/98, 16 pp., Univ. Franche-Comté, Besançon, 1999; MR1735371 (2001d:11109)]. Now a non-empty set T of finite primes is called a set of ramification for (K, S, p) when there exists an elementary cyclic p-extension L/K which is T -totally ramified and S -decomposed (i.e. unramified at the finite places outside T and such that every place in T totally ramifies and every place in S completely splits). In the paper under review the author is interested in minimal such sets T. By using the reflection techniques that he investigated in a previous article [J. Théor. Nombres Bordeaux 10 (1998), no. 2, 399–499 MR1828251 (2002g:11154)], he is able to characterize minimal ramification sets T in terms of decomposition subgoups of the places of T in the abelian T -unramified p-extension F(S)/K[ζ_p ] (from a review by Jean-François Jaulent ). NOTE: These questions are developped in our book "Class Field Theory" (Chap. V), Springer Monographs in Mathematics, Springer second corrected printing 2005.----- The whole PDF paper is available on: http://www.numdam.org/item?id=JTNB_2000__12_2_423_0 ..............................................................................................................................................................................................................................................FOR A COMPLETE VIEW OF MY PUBLICATIONS, PLEASE LOOK AT MY HOME PAGE: http://monsite.orange.fr/maths.g.mn.gras/