Relatively complete ordered fields without integer parts
Volume 179 / 2003
Fundamenta Mathematicae 179 (2003), 17-25 MSC: 12J15, 13J05, 54H13. DOI: 10.4064/fm179-1-2
We prove a convenient equivalent criterion for monotone completeness of ordered fields of generalized power series $[[F^G]]$ with exponents in a totally ordered Abelian group $G$ and coefficients in an ordered field $F$. This enables us to provide examples of such fields (monotone complete or otherwise) with or without integer parts, i.e. discrete subrings approximating each element within 1. We include a new and more straightforward proof that $[[F^G]]$ is always Scott complete. In contrast, the Puiseux series field with coefficients in $F$ always has proper dense field extensions.