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

18 pages