The $p$-rationality of $\mathbb{Q}\left(\sqrt{-(kp+m)}\right)$ and $\mathbb{Q}\left(\sqrt{p(p+1)}\right)$

Abstract

In this paper, we construct new families of imaginary and real quadratic fields that are $p$-rational. In the imaginary case, we prove that for any positive integer $k$ and any integer $m$, the imaginary quadratic field $\mathbb{Q}\left(\sqrt{-(kp+m)}\right)$ is $p$-rational for sufficiently large primes $p$. The proof relies on Louboutin's bound on the class numbers of imaginary quadratic fields. As a corollary, we recover the $p$-rationality of consecutive quadratic fields, a result due to Chattopadhyay, Laxmi and Saikia \cite{CLS}. In the real case, we give an explicit proof of the $p$-rationality of the real quadratic field $\mathbb{Q}\left(\sqrt{p(p+1)}\right)$ for any odd prime $p$, and obtain new pairs of real quadratic fields $\left(\mathbb{Q}\left(\sqrt{p(p-2)}\right),\mathbb{Q}\left(\sqrt{p(p-1)}\right)\right)$ and $\left(\mathbb{Q}\left(\sqrt{p(p+1)}\right),\mathbb{Q}\left(\sqrt{p(p+2)}\right)\right)$ for any prime $p>3$. We also construct new examples of $p$-rational triquadratic fields.