Now we fix an ultrafilter of $\mathbb{N}$ that contains the cofinite filter, consider a hyperreal field ${}^{*}\mathbb{R}$. Let $\varepsilon$ be a positive infinitesimal. We doubt that a power series field $\mathbb{R}((X))$ can be realized as a subquotient of ${}^{*}\mathbb{R}$.

In Measure, integration and elements of harmonic analysis on generalized loop spaces , Fesenko explained this in Remark 7.1. However, the meaning of some words appears unclear. For example, what does the phrase "the fraction field of approachable polynomials $\mathbb{R}[X]^{\rm ap}$'' mean?

If you know the answer or have any idea, please tell us.