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

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 .

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 .

DOI : 10.5802/jtnb.470
Viviani, Filippo 1

1 Universita’ degli studi di Roma Tor Vergata Dipartimento dimatematica via della ricerca scientifica 1 00133 Roma, Italy
@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},
     pages = {779--816},
     publisher = {Universit\'e Bordeaux 1},
     volume = {16},
     number = {3},
     year = {2004},
     doi = {10.5802/jtnb.470},
     zbl = {1075.11073},
     mrnumber = {2144967},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.470/}
}
TY  - JOUR
AU  - Viviani, Filippo
TI  - Ramification groups and Artin conductors of radical extensions of $\mathbb{Q}$
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2004
SP  - 779
EP  - 816
VL  - 16
IS  - 3
PB  - Université Bordeaux 1
UR  - http://www.numdam.org/articles/10.5802/jtnb.470/
DO  - 10.5802/jtnb.470
LA  - en
ID  - JTNB_2004__16_3_779_0
ER  - 
%0 Journal Article
%A Viviani, Filippo
%T Ramification groups and Artin conductors of radical extensions of $\mathbb{Q}$
%J Journal de théorie des nombres de Bordeaux
%D 2004
%P 779-816
%V 16
%N 3
%I Université Bordeaux 1
%U http://www.numdam.org/articles/10.5802/jtnb.470/
%R 10.5802/jtnb.470
%G en
%F 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, Tome 16 (2004) no. 3, pp. 779-816. doi : 10.5802/jtnb.470. http://www.numdam.org/articles/10.5802/jtnb.470/

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

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

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

[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 | Zbl

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

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

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

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

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

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

[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 | Zbl

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

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

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

[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 | Zbl

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

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

Cité par Sources :