  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.
