The $p$-smooth and $p$-golden partial fields

Abstract

Given a prime $p$, a $p$-smooth integer is an integer whose prime factors are all at most $p$. Let $S_p$ be the multiplicative subgroup of $\mathbb{Q}$ generated by $-1$ and the $p$-smooth integers. Define the $p$-smooth partial field as ${\mathbb{S}}_p=(\mathbb{Q},S_p)$. Let $g$ be the golden ratio $(1+\sqrt{5})/2$. Let $G_p$ to be the multiplicative subgroup of $\mathbb{R}$ generated by $g$, $-1$, and the $p$-smooth integers. Define the $p$-golden partial field as ${\mathbb{G}}_p=(\mathbb{R},G_p)$. The partial field ${\mathbb{S}}_2$ is actually the well-known dyadic partial field and ${\mathbb{S}}_3$ has sometimes been called the Gersonides partial field. We calculate the fundamental elements of ${\mathbb{S}}_5$, ${\mathbb{G}}_2$, ${\mathbb{G}}_3$, and ${\mathbb{G}}_5$.
Our proofs make use of the SageMath computational package.