PreprintCC BY 4.0arXiv
On the index of generalized trinomials over Valued fields
Abstract
Let ν be a Krull valuation of arbitrary rank on a field with valuation ring Rν, and let θ be a root of the irreducible polynomial F(x)=(xk+c)m−axn∈ Rν[x] and 1≤ n<km. We establish necessary and sufficient conditions for the integral closedness of Rν[θ], expressed explicitly in terms of the coefficients a, b, and the integers m, n, and k. In particular, when ν is the p-adic valuation on ℚ, our results yield criteria for determining the primes dividing the index [ℤK:ℤ[θ]], where K=ℚ(θ) and ℤK denotes the ring of integers of K.
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BibTeX
@misc{adithya2026index,
title = {{On the index of generalized trinomials over Valued fields}},
author = {V. ADITHYA},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29111},
eprint = {2609.29111},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29111},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the index of generalized trinomials over Valued fields
AU - V. ADITHYA
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29111
UR - https://arxiv.org/abs/2609.29111
AB - Let ν be a Krull valuation of arbitrary rank on a field with valuation ring R_ν, and let θ be a root of the irreducible polynomial F(x)=(x^k+c)^m−ax^n∈ R_ν[x] and 1≤ n<km. We establish necessary and sufficient conditions for the integral closedness of R_ν[θ], expressed explicitly in terms of the coefficients a, b, and the integers m, n, and k. In particular, when ν is the p-adic valuation on ℚ, our results yield criteria for determining the primes dividing the index [ℤ_K:ℤ[θ]], where K=ℚ(θ) and ℤ_K denotes the ring of integers of K.
ER -