Cyclic extensions of degree \(\ell\) over an \(\ell\)-regular number field. (Extensions cycliques de degré \(\ell\) de corps de nombres \(\ell\)-réguliers.) (French) Zbl 0834.11049

Soriano, Florence

J. Théor. Nombres Bordx. 6, No. 2, 407-420 (1994).

Let \(l\) be an odd prime. Any number field \(F\) is called \(l\)-regular if the \(l\)-part of the kernel of the regular symbols in \(K_2 (F)\) is trivial. Let \(K\) be an \(l\)-regular number field containing the \(l\)-th roots of unity. The main objective is to determine all cyclic extensions \(L/K\) of degree \(l\) such that \(L\) is also \(l\)-regular, and to classify these extensions according to the ramification index of the wild place in \(L/K\) and the \(l\)-adic value of the relative class number \(h_L/ h_K\). The case \(l=2\) has been dealt with earlier by R. Berger [Arch. Math. 59, 427-435 (1992; Zbl 0778.11062)]. The method is based on results of G. Gras and J.-F. Jaulent [Math. Z. 202, 343-365 (1989; Zbl 0704.11040)].

Reviewer: V.Ennola (Turku)

MSC:

11R70	\(K\)-theory of global fields
11R18	Cyclotomic extensions
11R20	Other abelian and metabelian extensions

Keywords:

\(K_ 2\) of a number field; \(l\)-regular number field; cyclic extensions

Citations:

Zbl 0778.11062; Zbl 0704.11040

References:

[1]	Berger, R., Class number parity and unit signature, Arch. Math.59 (1993), 427-435. · Zbl 0778.11062
[2]	Jaulent, J.-F., L’arithmétique des l-extensions (Thèse), Publ. Math. Fac. Sci. Besançon, Théor. Nombres (1984/ 1985, 1985/1986 (1986)), 13-43, 163-178. · Zbl 0601.12002
[3]	Jaulent, J.-F. & Nguyen, T.QUANG DO, Corps p-rationnels, corps p-réguliers, et ramification restreinte, J. Théor. Nombres Bordeaux5 (1994), 343-363. · Zbl 0957.11046
[4]	Gras, G. et Jaulent, J.-F., Sur les corps de nombres réguliers, Math.Z.202 (1989), 343-365. · Zbl 0704.11040
[5]	Serre, J.-P., Corps Locaux, Hermann, Paris (1968), 17-34, 211-238. · Zbl 0137.02601