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

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 نشر من قبل Christian Maire
 تاريخ النشر 2019
  مجال البحث
والبحث باللغة English
 تأليف Christian Maire




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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 Bockle-Guiraud-Kalyanswamy-Khare.



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