The p-rationality of ℚ(√(−(kp+m))) and ℚ(√(p(p+1)))

@misc{lin2026prationality,
  title = {{The $p$-rationality of $\mathbbQ\left(\sqrt-(kp+m)\right)$ and $\mathbbQ\left(\sqrtp(p+1)\right)$}},
  author = {Chen Lin and Xuejun Guo},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2607.21250},
  eprint = {2607.21250},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.21250},
}

TY  - UNPB
TI  - The p-rationality of ℚ(√(−(kp+m))) and ℚ(√(p(p+1)))
AU  - Chen Lin
AU  - Xuejun Guo
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2607.21250
UR  - https://arxiv.org/abs/2607.21250
AB  - 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 \cite{CLS}. 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.
ER  -