On simple extensions generated by key polynomials
Abstract
Let Kh be a Henselization of a valued field (K,v), w a valuation-transcendental extension of v to K[x], and ϕ a key polynomial for w. In this paper, we prove that if ϕ is irreducible over Kh, then the simple extension L|K, generated by a root of any key polynomial for w, is unibranched and its defect is independent of the choice of key polynomial. As a consequence, we generalize some well-known results for Henselian valued fields to arbitrary valued fields. Moreover, we provide some criteria for L|K to be unibranched and defectless in terms of Mac Lane-Vaquié chains, complete sequences of abstract key polynomials and saturated distinguished chains. As an application, we generalize a classical result of Ore about the existence of p-regular generators for number fields.
Related
- Extensions of Restricted Simple Modules for Restricted Lie Algebras of Cartan Type
- Infinitely many finite simple groups of Lie type are not generated invariably by two elements of prime order
- Algebraic Vector Fields of Connected Semi-simple Matrix Groups are 3-generated
- Orthogonal polynomials and the centres of the universal central extensions of superelliptic Krichever--Novikov algebras
- Density of polynomials whose Aron--Berner extensions attain their norms
BibTeX
@misc{dwivedi2026simple,
title = {{On simple extensions generated by key polynomials}},
author = {Nikita Dwivedi and Anuj Bishnoi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15450},
eprint = {2609.15450},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15450},
}
RIS
TY - UNPB
TI - On simple extensions generated by key polynomials
AU - Nikita Dwivedi
AU - Anuj Bishnoi
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15450
UR - https://arxiv.org/abs/2609.15450
AB - Let K^h be a Henselization of a valued field (K,v), w a valuation-transcendental extension of v to K[x], and ϕ a key polynomial for w. In this paper, we prove that if ϕ is irreducible over K^h, then the simple extension L|K, generated by a root of any key polynomial for w, is unibranched and its defect is independent of the choice of key polynomial. As a consequence, we generalize some well-known results for Henselian valued fields to arbitrary valued fields. Moreover, we provide some criteria for L|K to be unibranched and defectless in terms of Mac Lane-Vaquié chains, complete sequences of abstract key polynomials and saturated distinguished chains. As an application, we generalize a classical result of Ore about the existence of p-regular generators for number fields.
ER -