Triquadratic p-rational fields

Abstract

In his work about Galois representations, R. Greenberg conjectured the existence, for any odd prime p and any positive integer t, of a multiquadratic p-rational number field of degree 2t.In this article, we prove that there exists infinitely many primes p such that the triquadratic field Q(p(p+2),p(p−2),−1) is p-rational.To do this, we use an analytic result providing us with infinitely many primes p such that p−2 and p+2 simultaneously have large square factors. Therefore the related imaginary quadratic subfields Q(−p(p+2)), Q(−p(p−2)) and Q(−(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.