An introduction to oddly tame number fields.
Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 3, pp. 711-717.

It follows from generalities of quadratic forms that the spinor class of the integral trace of a number field determines the signature and the discriminant of the field. In this paper we define a family of number fields, that contains among others all odd degree Galois tame number fields, for which the converse is true. In other words, for a number field K in such family we prove that the spinor class of the integral trace carries no more information about K than the discriminant and the signature do.

Il résulte des généralités sur les formes quadratiques que la classe spinorielle de la trace intégrale d’un corps de nombres détermine la signature et le discriminant du corps. Dans cet article, nous définissons une famille de corps de nombres, qui contient, entre autres, tous les corps galoisiens de degré impair modérément ramifiés, pour lesquels la réciproque est vraie. Autrement dit, pour un corps de nombres K de cette famille, on montre que la classe spinorielle de la trace intégrale ne contient pas d’informations sur K autres que celles qui sont fournies par le discriminant et la signature.

Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1140
Classification: 11R04,  11S15
Keywords: Arithmetic invariants, tame fields, arithmetic equivalence, trace forms.
Guillermo Mantilla-Soler 1, 2

1 Department of Mathematics Fundación Universitaria Konrad Lorenz Bogotá, Colombia
2 Department of Mathematics and Systems Analysis Aalto University Helsinki, Finland
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Guillermo Mantilla-Soler. An introduction to oddly tame number fields.. Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 3, pp. 711-717. doi : 10.5802/jtnb.1140. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1140/

[1] Wilmar Bolaños; Guillermo Mantilla-Soler The trace form over cyclic number fields (2019) (https://arxiv.org/abs/1904.10080v2, to appear in Can. J. Math.)

[2] J. W. S. Cassels Rational quadratic forms, Dover Publications, 2008

[3] John Jones; David Roberts A data base of number fields, LMS J. Comput. Math., Volume 17 (2014) no. 1, pp. 595-618 | Article | Zbl: 1360.11121

[4] Guillermo Mantilla-Soler On the arithmetic determination of the trace, J. Algebra, Volume 444 (2015), pp. 272-283 | Article | MR: 3406177 | Zbl: 1326.11066

[5] Guillermo Mantilla-Soler The Spinor Genus of the integral trace, Trans. Am. Math. Soc., Volume 369 (2017) no. 3, pp. 1547-1577 | MR: 3581214 | Zbl: 1410.11031

[6] Guillermo Mantilla-Soler The (α,β)-ramification invariants of a number field (2019) (https://arxiv.org/abs/1906.04254, to appear in Mathematica Slovaca)

[7] Guillermo Mantilla-Soler; Carlos Rivera-Guaca An introduction to Casimir pairings and some arithmetic applications (2019) (https://arxiv.org/abs/1812.03133v3)

[8] Jean-Pierre Serre Local fields, Graduate Texts in Mathematics, 67, Springer, 1979, viii+241 pages | Zbl: 0423.12016

[9] Olga Taussky The discriminant matrices of an algebraic number field, J. Lond. Math. Soc., Volume 43 (1968), pp. 152-154 | Article | MR: 228473

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