2022/08/08 by Jaitra Chattopadhyay, Chattopadhyay, Jaitra, H Laxmi +3
Arts and Humanities · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Historical Studies and Socio-cultural Analysis #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2208.04214
openalex publication_date 2022/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 2016, in the work related to Galois representations, Greenberg conjectured the existence of multi-quadratic p-rational number fields of degree 2t for any odd prime number p and any integer t ≥ 1. Using the criteria provided by him to check p-rationality for abelian number fields, certain infinite families of quadratic, biquadratic and triquadratic p-rational fields have been shown to exist in recent years. In this article, for any integer k ≥ 1, we build upon the existing work and prove the existence of infinitely many prime numbers p for which the imaginary quadratic fields ℚ(√(-(p - 1))),…,ℚ(√(-(p - k))) and ℚ(√(-p(p - 1))),…, ℚ(√(-p(p - k))) are all p-rational. This can be construed as analogous results in the spirit of Iizuka's conjecture on the divisibility of class numbers of consecutive quadratic fields. We also address a similar question of p-rationality for two consecutive real quadratic fields by proving the existence of infinitely many p-rational fields of the form ℚ(√p2 + 1) and ℚ(√p2 + 2). The result for imaginary quadratic fields is accomplished by producing infinitely many primes for which the corresponding consecutive discriminants have large square divisors and the same for real quadratic fields is proven using a result of Heath-Brown on the density of square-free values of polynomials at prime arguments.