2021/03/29 by Julien Koperecz, Koperecz, Julien
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 · doi:10.48550/arxiv.2103.15648
openalex publication_date 2021/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In his work about Galois representations, Greenberg conjectured the existence, for any odd prime p and any positive integer t, of a multiquadratic p-rational number field of degree 2 t. In this article, we prove that there exists infinitely many primes p such that the triquadratic field Q(p(p + 2), p(p -- 2), i) is p-rational. To do this, we use an analytic result, proved apart in section \S4, providing us with infinitely many prime numbers p such that p + 2 et p -- 2 have ''big'' square factors. Therefore the related imaginary quadratic subfields Q(i √ p + 2), Q(i √ p -- 2) and Q(i (p + 2)(p -- 2)) have ''small'' discriminants for infinitely many primes p. In the spirit of Brauer-Siegel estimates, it proves that the class numbers of these imaginary quadratic fields are relatively prime to p, and so prove their p-rationality.