Ramification invariants of pure extensions
Abstract
We study ramification invariants of finite separable extensions of valued fields that are pure on a primitive element. In the immediate unibranched case, we prove that the extended trace ideal is generated by minimal-polynomial derivatives of integral primitive elements and that the Dedekind different is its divisorial closure. For Galois extensions, we express this derivative ideal as a product of subgroup ramification ideals. These formulas extend prime-degree calculations to pure extensions of higher degree. We compute the differential module, traces, differents, ramification ideals, and character Swan ideals of a pure immediate elementary abelian extension of degree p2. A totally ramified example of the same degree gives a comparison with the defectless case.
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BibTeX
@misc{novacoski2026ramification,
title = {{Ramification invariants of pure extensions}},
author = {Josnei Novacoski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03909},
eprint = {2610.03909},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03909},
}
RIS
TY - UNPB
TI - Ramification invariants of pure extensions
AU - Josnei Novacoski
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03909
UR - https://arxiv.org/abs/2610.03909
AB - We study ramification invariants of finite separable extensions of valued fields that are pure on a primitive element. In the immediate unibranched case, we prove that the extended trace ideal is generated by minimal-polynomial derivatives of integral primitive elements and that the Dedekind different is its divisorial closure. For Galois extensions, we express this derivative ideal as a product of subgroup ramification ideals. These formulas extend prime-degree calculations to pure extensions of higher degree. We compute the differential module, traces, differents, ramification ideals, and character Swan ideals of a pure immediate elementary abelian extension of degree p^2. A totally ramified example of the same degree gives a comparison with the defectless case.
ER -