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arXiv · 1903.11271

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

Christian Maire·Marine Rougnant
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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 K$/\mathbb{Q}$ a real quadratic extension or an imaginary $S_3$-extension, if the generalized $abc$-conjecture holds in K, then there exist at least $c\,\log X$ prime numbers $p \leq X$ for which K is $p$-rational, here $c$ is some nonzero constant depending on K. The real quadratic case was recently suggested by B\"ockle-Guiraud-Kalyanswamy-Khare.

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Christian Maire, Marine Rougnant. 2019-03-27. A note on $p$-rational fields and the abc-conjecture. https://arxiv.org/abs/1903.11271

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