no. 2
Extensions cycliques de degré ℓ de corps de nombres ℓ-réguliers
Journal de théorie des nombres de Bordeaux, Volume 6 (1994) no. 2, pp. 407-420.
corrected-by Erratum à l’article « Extensions cycliques de degré ℓ de corps de nombres ℓ-réguliers »

We determine all cyclic extensions L of prime degree ℓ over a ℓ-regular number field K containing the ℓ-roots of unity which are also l-regular. We classify these extensions according to the ramification index of the wild place ℒ in L/K and to the ℓ-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 ℓ 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.

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     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},
     zbl = {0834.11049},
     mrnumber = {1360653},
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     url = {http://www.numdam.org/item/JTNB_1994__6_2_407_0/}
}
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Soriano, Florence. Extensions cycliques de degré $\ell $ de corps de nombres $\ell $-réguliers. Journal de théorie des nombres de Bordeaux, Volume 6 (1994) no. 2, pp. 407-420. http://www.numdam.org/item/JTNB_1994__6_2_407_0/

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

[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

[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 | Zbl

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

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