A Choice-free look at algebraic extensions of valuations
Cédric Aïd
TL;DR
The paper develops a choice-free framework for extending a valuation $v$ from a base field $K$ to a finite algebraic extension $L$ by establishing a bijection between valuation extensions and maximal ideals of the relative integral closure $R$ of $\\mathcal{O}_v$ in $L$. It shows that $R/J(R)$ is a finite-dimensional, reduced $\\kappa_v$-algebra that decomposes into a product of field factors, yielding an explicit, choice-free description of all extensions and yielding weak approximation-type results. For finite extensions, the correspondence via localization $R_I$ provides concrete extensions with explicit residue fields, and an elementary proof of the fundamental inequality $\\sum_i e(w_i|v) f(w_i|v) \\\le [L:K]$ is given. Overall, this approach avoids Zorn’s lemma and related choice principles, while still guaranteeing the existence of valuation extensions and clarifying their structural constraints.
Abstract
In this paper, we study extensions of valuations over algebraic field extensions without the use of the Axiom of Choice. We show a bijection between the extensions of a valuation and the maximal ideals of the relative integral closure of its valuation ring. In the case of a finite extension, we show that these maximal ideals exist. We conclude with an elementary proof of the fundamental inequality.