Extensions cycliques de degré de corps de nombres -réguliers
Journal de Théorie des Nombres de Bordeaux, Tome 6 (1994) no. 2, pp. 407-420.

We determine all cyclic extensions L of prime degree over a -regular number field K containing the -roots of unity which are also l-regular. We classify these extensions according to the ramification index of the wild place in L/K and to the -valuation of the relative class number h L/K (which is the quotient h L /h K of the ordinary class numbers of L and K). We study the case where the is odd prime, since the even case was studien by R. Berger. Our genus theory methods rely essentially on G. Gras and J.-F Jaulent’s results.

@article{JTNB_1994__6_2_407_0,
     author = {Soriano, Florence},
     title = {Extensions cycliques de degr\'e $\ell $ de corps de nombres $\ell $-r\'eguliers},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {407--420},
     publisher = {Universit\'e Bordeaux I},
     volume = {6},
     number = {2},
     year = {1994},
     doi = {10.5802/jtnb.122},
     zbl = {0834.11049},
     mrnumber = {1360653},
     language = {fr},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.122/}
}
Florence Soriano. Extensions cycliques de degré $\ell $ de corps de nombres $\ell $-réguliers. Journal de Théorie des Nombres de Bordeaux, Tome 6 (1994) no. 2, pp. 407-420. doi : 10.5802/jtnb.122. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.122/

[B] R. Berger, Class number parity and unit signature, Arch. Math. 59 (1993), 427-435. | MR 1185340 | Zbl 0778.11062

[J] J.-F. Jaulent, 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

[JN] J.-F. Jaulent & T. Nguyen QUANG DO, Corps p-rationnels, corps p-réguliers, et ramification restreinte, J. Théor. Nombres Bordeaux 5 (1994), 343-363. | Numdam | MR 1265910 | Zbl 0957.11046

[GJ] G. Gras et J.-F. Jaulent, Sur les corps de nombres réguliers, Math.Z.202 (1989), 343-365. | MR 1017575 | Zbl 0704.11040

[S] J.-P. Serre, Corps Locaux, Hermann, Paris (1968), 17-34, 211-238. | MR 354618 | Zbl 0137.02601