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


Authors: Christian Maire and Marine Rougnant
Journal: Proc. Amer. Math. Soc. 148 (2020), 3263-3271
MSC (2010): Primary 11R37, 11R23
DOI: https://doi.org/10.1090/proc/14983
Published electronically: April 14, 2020
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Abstract: 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 $ \operatorname {K}/\mathbb{Q}$ a real quadratic extension or an imaginary $ S_3$-extension, if the generalized $ abc$-conjecture holds in  $ \operatorname {K}$, then there exist at least $ c\ \log X$ prime numbers $ p \leq X$ for which  $ \operatorname {K}$ is $ p$-rational; here $ c$ is some nonzero constant depending on  $ \operatorname {K}$. The real quadratic case was recently suggested by Böckle-Guiraud-Kalyanswamy-Khare.


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Christian Maire
Affiliation: FEMTO-ST Institute, Université Bourgogne Franche-Comté, 15B Avenue des Montboucons, 25030 Besançon Cedex, France
Email: christian.maire@univ-fcomte.fr

Marine Rougnant
Affiliation: Laboratoire de Mathématiques de Besançon, Université Bourgogne Franche-Comté, UFR Sciences et Techniques, 16 route de Gray, 25030 Besançon Cedex, France
Email: marine.rougnant@univ-fcomte.fr

DOI: https://doi.org/10.1090/proc/14983
Keywords: $p$-rationals fields, $abc$-conjecture
Received by editor(s): March 27, 2019
Received by editor(s) in revised form: June 23, 2019, and December 15, 2019
Published electronically: April 14, 2020
Additional Notes: The authors were supported in part by the ANR project FLAIR (ANR-17-CE40-0012).
The first author was also supported by the EIPHI Graduate School (ANR-17-EURE-0002)
Communicated by: Rachel Pries
Article copyright: © Copyright 2020 American Mathematical Society