I'm looking for a reference on splitting fields for 'generalized polynomials' over non-Archimedean fields with exponents in a non-Archimedean discretely ordered value class.
To be precise, let $\mathbb{Q}^*(T^{\omega^{\omega^\alpha}})$ be the monoid algebra whose base field is a nonstandard model of the rationals $\mathbb{Q}^*$ and whose monoid is some $\delta$-number $\omega^{\omega^\alpha}$ under natural addition, for $\alpha\in O_n$ fixed. Then $\mathbb{Q}^*(T^{\omega^{\omega^\alpha}})$ is a set of 'generalized polynomials' in the appropriate sense, and I'm curious about field extensions like $\mathbb{Q}^*[\sqrt{x}]$ where $x\in\mathbb{Q}^*$ is a nonstandard (infinite) rational number, which is the splitting field of $T^2-x\in\mathbb{Q}^*(T^{\omega^{\omega^\alpha}})$.
