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arXiv · 2007.04864

Multi-quadratic $p$-rational Number Fields

Youssef Benmerieme·Abbas Movahhedi
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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 $GL_n(\mathbf{Z_p})$, for $n =4$ and $n =5$ and at least for all the primes $p <192.699.943$.

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Youssef Benmerieme, Abbas Movahhedi. 2020-07-09. Multi-quadratic $p$-rational Number Fields. https://arxiv.org/abs/2007.04864

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