Triquadratic p-Rational Fields

TL;DR

This paper proves the existence of infinitely many primes p for which certain triquadratic fields are p-rational, using analytic methods to analyze the properties of related quadratic subfields and their class numbers.

Contribution

It establishes the infinite occurrence of p-rational triquadratic fields for infinitely many primes p, confirming a conjecture related to Galois representations and p-rationality.

Findings

Infinitely many primes p make the field Q(p(p+2), p(p-2), i) p-rational.

The related quadratic subfields have small discriminants for infinitely many p.

Class numbers of these quadratic fields are relatively prime to p, ensuring p-rationality.

Abstract

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 $\sqrt$ p + 2), Q(i $\sqrt$ 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.