2026/07/23 by Chen Lin, Xuejun Guo
Mathematics · #math.NT
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 ℚ(√(-(kp+m))) 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 \citeCLS. In the real case, we give an explicit proof of the p-rationality of the real quadratic field ℚ(√(p(p+1))) for any odd prime p, and obtain new pairs of real quadratic fields (ℚ(√(p(p-2))),ℚ(√(p(p-1)))) and (ℚ(√(p(p+1))),ℚ(√(p(p+2)))) for any prime p>3. We also construct new examples of p-rational triquadratic fields.