Depth of extensions of valuations
Josnei Novacoski, Enric Nart
TL;DR
This work defines and analyzes the depth of simple extensions $(L/K,v)$ of valued fields via Mac Lane–Vaquié chains, connecting depth to Okutsu sequences and ultrametric balls. It develops residual polynomial operators and graded-algebra tools to study augmentations and residue degrees, and proves a defectless (Henselian) depth bound of $2$, generalizing Ore’s $p$-regular generator result; it also provides depth-one characterizations, explicit depth-3 and depth-two tame examples, and a non-Henselian case illustrating depth behavior beyond henselization. The results yield practical criteria and methods for computing depth, illuminating how valuation-theoretic structure, defect, and ramification influence generator behavior and, by extension, prime-ideal decompositions in number-theoretic contexts. Overall, the paper advances the valuation-theoretic understanding of extension depth and provides a framework to classify and compute depth across diverse extensions.
Abstract
In this paper we develop the theory of the depth of a simple algebraic extension of valued fields $(L/K,v)$. This is defined as the minimal number of augmentations appearing in some Mac Lane-Vaquié chain for the valuation on $K[x]$ determined by the choice of some generator of the extension. In the defectless and unibranched case, this concept leads to a generalization of a classical result of Ore about the existence of $p$-regular generators for number fields. Also, we find what valuation-theoretic conditions characterize the extensions having depth one.