Ramification groups and Artin conductors of radical extensions of
Journal de théorie des nombres de Bordeaux, Volume 16 (2004) no. 3, p. 779-816

We study the ramification properties of the extensions (ζ m ,a m)/ under the hypothesis that m is odd and if pm than either pv p (a) or p v p (m) v p (a) (v p (a) and v p (m) are the exponents with which p divides a and m). In particular we determine the higher ramification groups of the completed extensions and the Artin conductors of the characters of their Galois group. As an application, we give formulas for the p-adique valuation of the discriminant of the studied global extensions with m=p r .

Nous étudions les propriétés de ramification des extensions (ζ m ,a m)/ sous l’hypothèse que m est impair et si pm, ou bien pv p (a) ou bien p v p (m) v p (a) (v p (m) et v p (a) sont les exposants avec lesquels p divise a et m). En particulier, nous déterminons les groupes de ramification supérieurs des extensions complétées et les conducteurs d’Artin des caractères de leur groupe de Galois. A titre d’application, nous donnons des formules pour la valuation p-adique du discriminant des extensions globales considérées avec m=p r .

@article{JTNB_2004__16_3_779_0,
     author = {Viviani, Filippo},
     title = {Ramification groups and Artin conductors of radical extensions of $\mathbb{Q}$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {16},
     number = {3},
     year = {2004},
     pages = {779-816},
     doi = {10.5802/jtnb.470},
     mrnumber = {2144967},
     zbl = {1075.11073},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2004__16_3_779_0}
}
Viviani, Filippo. Ramification groups and Artin conductors of radical extensions of $\mathbb{Q}$. Journal de théorie des nombres de Bordeaux, Volume 16 (2004) no. 3, pp. 779-816. doi : 10.5802/jtnb.470. http://www.numdam.org/item/JTNB_2004__16_3_779_0/

[1] M. Acosta, W. Y. Velez, The lattice of subfields of radicals extensions. Journal of Number theory 15 (1982), 388–405. | MR 680540 | Zbl 0493.12027

[2] J.W.S. Cassels, A. Fröhlich, Algebraic number theory. Academic press: London, 1967. | MR 215665 | Zbl 0153.07403

[3] H. Hasse, Number theory. Springer-Verlag: New York, 1980. | MR 562104 | Zbl 0423.12002

[4] E.T. Jacobson, W. Y. Velez, The Galois group of a radical extension of the rationals. Manuscripta Math. 67 no. 3 (1990), 271–284. | MR 1046989 | Zbl 0717.12002

[5] K. Komatsu, An integral bases of the algebraic number field (a l,1 l). J. Reine Angew. Math. 288 (1976), 152–153. | MR 422201 | Zbl 0335.12016

[6] S. Lang, Algebra, revised third edition. Springer-Verlag: New York, 2002. | MR 1878556 | Zbl 0984.00001

[7] H. B. Mann, W. Y. Velez, Prime ideal decomposition in F(μ m). Monatsh. Math. 81 (1976), 131–139. | MR 399043 | Zbl 0324.12003

[8] B. Mora, W. Y. Velez, Some results on radical extensions. J. of Algebra 162 (1993), 295–301. | MR 1254775 | Zbl 0798.12005

[9] A. Schinzel, Abelian binomials, power residues and exponential congruences. Acta Arith. 32 (1977), 245–274. | MR 429819 | Zbl 0409.12029

[10] J.P. Serre, Local fields. Springer-Verlag: New York, 1979. | MR 554237 | Zbl 0423.12016

[11] W. Y. Velez, A generalization of Schinzel’s theorem on radical extensions of fields and an application. Acta Arith. 51 no. 2 (1988), 119–130. | MR 975106 | Zbl 0662.12025

[12] W.Y. Velez, On normal binomials. Acta Arith. 36 (1980), 113–124. | MR 581910 | Zbl 0487.12013

[13] W. Y. Velez, Prime ideal decomposition in F(μ p). Pacific Journal of mathematics 75 no. 2 (1978), 589–600. | MR 506215 | Zbl 0344.12002

[14] W. Y. Velez, Several results on radical extensions. Arch. Math. (Basel) 45 no. 4 (1985), 342–349. | MR 810252 | Zbl 0589.12018

[15] W. Y. Velez, The factorization of p in (a p k ) and the genus field of (a n). Tokyo J. Math. 11 no. 1 (1988), 1–19. | MR 947943 | Zbl 0664.12003

[16] J. Westlund, On the fundamental number of the algebraic number field K(m p). Trans. Amer. Math. Soc. 11 (1910), 388–392. | MR 1500870

[17] J. Wójcik, Contributions to the theory of Kummer extensions. Acta Arith. 40 (1982), 155–174. | MR 649116 | Zbl 0491.12002