Triquadratic p-Rational Fields

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 fieldTo do this, we use an analytic result, proved apart in section §4, providing us with infinitely many prime numbers p such that p + 2 et p − 2 have "big" square factors. Therefore the related imaginary quadratic subfieldshave "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.