Greenberg's p-rationality conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a number field $K$ and an odd prime $p$, let $K_S$ be the maximal pro-$p$ extension of $K$ unramified at every finite prime not dividing $p$. The field is $p$-rational if $\operatorname{Gal}(K_S/K)$ is a free pro-$p$ group. A pro-$p$ group is an inverse limit of finite $p$-groups, and freeness is the universal mapping property in the category of pro-$p$ groups. A multiquadratic extension of rank $t$ is a finite Galois extension with Galois group $(\mathbb Z/2\mathbb Z)^t$.

Statement recorded in the supplied source.

For every odd prime $p$ and every integer $t\ge1$, does there exist a $p$-rational number field $K$, Galois over $\mathbb Q$, with $\operatorname{Gal}(K/\mathbb Q)\cong(\mathbb Z/2\mathbb Z)^t$? [S2]

Short English statement

Do multiquadratic fields of arbitrarily large prescribed rank exist that are p-rational, for each odd prime?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400181 (AMR-093-0181). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Y. Benmerieme and A. Movahhedi, Multi-quadratic p-rational Number Fields, arXiv:2007.04864v1 (2020), Section 1, Greenberg conjecture, p. 1; Section 2, definition of p-rationality, pp. 2--3. https://arxiv.org/pdf/2007.04864.