Multi-quadratic $p$-rational Number Fields

TL;DR

This paper proves the existence of infinitely many $p$-rational quadratic fields for odd primes and constructs explicit bi-quadratic $p$-rational fields, also exploring Galois extensions with large Galois groups.

Contribution

It establishes the existence of infinitely many $p$-rational quadratic fields and constructs explicit examples, extending the understanding of $p$-rationality in number fields.

Findings

Infinitely many real quadratic $p$-rational fields exist for each odd prime $p$.

Explicit constructions of imaginary and real bi-quadratic $p$-rational fields are provided.

Galois extensions with Galois group isomorphic to open subgroups of $GL_n(\mathbf{Z}_p)$ are shown to exist for certain primes.

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 $p <192.699.943$.