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Pré-Publication, Document De Travail Année : 2019

A note on $p$-rational fields and the abc-conjecture

Résumé

In this short note we confirm the relation between the generalized abc-conjecture and the p-rationality of number fields. Namely, we prove that given K{Q a real quadratic extension or an imaginary dihedral extension of degree 6, if the generalized abc-conjecture holds in K, then there exist at least c logX prime numbers p ď X for which K is p-rational, here c is some nonzero constant depending on K. The real quadratic case was recently suggested by Böckle-Guiraud-Kalyanswamy-Khare.
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Dates et versions

hal-02077680 , version 1 (23-03-2019)
hal-02077680 , version 2 (22-06-2019)

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Christian Maire, Marine Rougnant. A note on $p$-rational fields and the abc-conjecture. 2019. ⟨hal-02077680v2⟩
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