Triquadratic p-rational fields

In his work about Galois representations, R. Greenberg conjectured the existence, for any odd prime p and positive integer t, of a multiquadratic p-rational number field degree 2t. this article, we prove that there exists infinitely many primes such triquadratic Q(p(p+2),p(p−2),−1) is p-rational. To do this, use an analytic result providing us with p−2 p+2 simultaneously have large square factors. Therefore related imaginary quadratic subfields Q(−p(p+2)), Q(−p(p−2)) Q(−(p+2)(p−2)) small discriminants p. spirit Brauer-Siegel estimates, it proves class numbers these fields are relatively to p, so their p-rationality.