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 $p <192.699.943$.

中文翻译：

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多二次 p 有理数域

对于每个奇素数 $p$，我们证明存在无穷多个 $p$-有理的实二次域。还为每个素数 $p$ 给出了显式的虚和实双二次 $p$-有理场。使用格林伯格最近开发的方法，我们推导出 $\mathbf{Q}$ 的伽罗瓦扩展的存在性，伽罗瓦群同构为 $GL_n(\mathbf{Z_p})$ 的开子群，对于 $n =4$ 和$n = 5$ 并且至少对于所有素数 $p <192.699.943$。