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arXiv · 2503.00850

Depth of extensions of valuations

Josnei Novacoski·Enric Nart
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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-Vaqui\'e 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.

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Josnei Novacoski, Enric Nart. 2025-03-02. Depth of extensions of valuations. https://arxiv.org/abs/2503.00850

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