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

## TL;DR

This paper explores the connection between the generalized abc-conjecture and p-rationality of certain number fields, proving that if the conjecture holds, then many primes induce p-rationality in these fields.

## Contribution

It establishes a conditional link between the generalized abc-conjecture and the abundance of primes making specific number fields p-rational, extending recent suggestions in the field.

## Key findings

- Conditional proof of p-rational primes in real quadratic fields
- Extension of results to imaginary S_3-extensions
- Quantitative lower bound on the number of p-rational primes

## 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.