Approximating DVRs by elements of bounded ramification
Abstract
Let V be a DVR with quotient field K and perfect residue field, v be the valuation on K associated with V, K̂ be the completion of K, and 𝕂 be the completion of an algebraic closure K̂̅ of K̂. We show that a DVR of the rational function field K(X) which is a residually algebraic extension of V is necessarily of the form Vα={ϕ∈ K(X)| v(ϕ(α))≥0}, for an element α of 𝕂 transcendental over K, and that α is algebraic over K̂ if and only if the residue field extension is finite. Not every such Vα is a DVR, however, and we characterize the α∈𝕂 for which Vα is a DVR: they are the elements which can be approximated by algebraic elements in K̂̅ with bounded ramification indexes. Combining the two results, we obtain a complete description of the extensions of V to K(X) which are DVRs and residually algebraic over V, together with a criterion for each of the two cases to occur. The proofs rest on a bound for the ramification index in a compositum, valid with no tameness assumption and under a separability hypothesis on one residue field extension only; we show that the inequality cannot be improved to a divisibility and that this hypothesis cannot be dropped. We also show that the hypothesis of discreteness cannot be omitted. Furthermore, we show that the set of α∈𝕂 for which Vα is a DVR is a subfield of 𝕂, which sits properly between K̂̅ and 𝕂, and corresponds to those elements α for which the value group of K̂(α) is discrete.
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BibTeX
@misc{chang2026approximating,
title = {{Approximating DVRs by elements of bounded ramification}},
author = {Gyu Whan Chang and Giulio Peruginelli},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30483},
eprint = {2608.30483},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30483},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Approximating DVRs by elements of bounded ramification
AU - Gyu Whan Chang
AU - Giulio Peruginelli
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30483
UR - https://arxiv.org/abs/2608.30483
AB - Let V be a DVR with quotient field K and perfect residue field, v be the valuation on K associated with V, K̂ be the completion of K, and 𝕂 be the completion of an algebraic closure K̂̅ of K̂. We show that a DVR of the rational function field K(X) which is a residually algebraic extension of V is necessarily of the form V_α={ϕ∈ K(X)| v(ϕ(α))≥0}, for an element α of 𝕂 transcendental over K, and that α is algebraic over K̂ if and only if the residue field extension is finite. Not every such V_α is a DVR, however, and we characterize the α∈𝕂 for which V_α is a DVR: they are the elements which can be approximated by algebraic elements in K̂̅ with bounded ramification indexes. Combining the two results, we obtain a complete description of the extensions of V to K(X) which are DVRs and residually algebraic over V, together with a criterion for each of the two cases to occur. The proofs rest on a bound for the ramification index in a compositum, valid with no tameness assumption and under a separability hypothesis on one residue field extension only; we show that the inequality cannot be improved to a divisibility and that this hypothesis cannot be dropped. We also show that the hypothesis of discreteness cannot be omitted. Furthermore, we show that the set of α∈𝕂 for which V_α is a DVR is a subfield of 𝕂, which sits properly between K̂̅ and 𝕂, and corresponds to those elements α for which the value group of K̂(α) is discrete.
ER -