  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 $\mathbf{Q}$ with Galois group isomorphic to an open subgroup of
$GL_n(\mathbf{Z_p})$, for $n =4$ and $n =5$ and at least for all the primes $p
<192.699.943$.
