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.