On p-rationality of number fields. Applications – PARI/GP programs
Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 2 (2019), pp. 29-51.

Let K be a number field. We prove that its ray class group modulo p 2 (resp. 8) if p>2 (resp. p=2) characterizes its p-rationality. Then we give two short and very fast PARI Programs (Sections 3.1, 3.2) testing if K (defined by an irreducible monic polynomial) is p-rational or not. For quadratic fields we verify some densities of 3-rational fields related to Cohen–Lenstra–Martinet ones and analyse Greenberg’s conjecture on the existence of p-rational fields with Galois groups (/2) t needed for the construction of some Galois representations with open image. We give examples for p=3, t=5 and t=6 (Sections 5.1, 5.2) and illustrate other approaches (Pitoun–Varescon, Barbulescu–Ray). We conclude about the existence of imaginary quadratic fields, p-rational for all p2 (Angelakis–Stevenhagen study on the concept of “minimal absolute abelian Galois group”) which may enlighten a conjecture of p-rationality (Hajir–Maire) giving large Iwasawa μ-invariants of some uniform pro-p-groups. All programs (in “verbatim”) can be used by the reader by simply copied and pasted.

Soit K un corps de nombres. Nous montrons que son corps de classes de rayon modulo p 2 (resp. 8) si p>2 (resp. p=2) caractérise sa p-rationalité. Puis nous donnons deux programmes PARI (Sections 3.1, 3.2) très courts et rapides testant si K (défini par un polynôme irréductible unitaire) est p-rationnel ou non. Pour les corps quadratiques nous vérifions certaines densités de corps 3-rationnels en relation avec celles de Cohen–Lenstra–Martinet et nous analysons la conjecture de Greenberg sur l’existence de corps p-rationnels de groupes de Galois (/2) t nécessaires pour la construction de certaines représentations galoisiennes d’image ouverte. Nous donnons des exemples pour p=3, t=5 et t=6 (Sections 5.1, 5.2) et illustrons d’autres approches (Pitoun–Varescon, Barbulescu–Ray). Nous concluons sur l’existence de corps quadratiques imaginaires p-rationnels pour tout p2 (étude de Angelakis–Stevenhagen sur le concept de “groupe de Galois abélien absolu minimal”) qui peut éclairer une conjecture de p-rationalité (Hajir–Maire) donnant de grands invariants μ d’Iwasawa relatifs à certains pro-p-groupes uniformes. Tous les programmes (en “verbatim”) sont utilisables par le lecteur par simple copié-collé.

Published online:
DOI: 10.5802/pmb.35
Classification: 11R04, 11R37, 11R11, 08-04
Keywords: $p$-rational fields, class field theory, abelian $p$-ramification, PARI/GP programs, quadratic fields, Greenberg’s conjecture on representations
Gras, Georges 1

1 Villa la Gardette chemin Château Gagnière 38520 Le Bourg d’Oisans, France
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Gras, Georges. On $p$-rationality of number fields. Applications – PARI/GP programs. Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 2 (2019), pp. 29-51. doi : 10.5802/pmb.35. http://www.numdam.org/articles/10.5802/pmb.35/

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