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

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{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 d 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{o}ckle-Guiraud-Kalyanswamy-Khare.