Multi-quadratic $p$-rational Number Fields

Abstract

For each odd prime p, we prove the existence of infinitely many real quadratic fields which are p-rational. Explicit imaginary and real bi-quadratic p-rational fields are also given for each prime p. Using a recent method developed by Greenberg, we deduce the existence of Galois extensions of Q with Galois group isomorphic to an open subgroup of GLn(Zp), for n = 4 and n = 5 and at least for all the primes p < 192.699.943.