Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\nu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation> be a Krull valuation of arbitrary rank on a field with valuation ring <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$R_\nu $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mi>ν</mi> </msub> </math></EquationSource> </InlineEquation>, and let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation> be a root of the irreducible polynomial <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$F(x) = x^{n - km}(x^k + a)^m + b,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>x</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>k</mi> <mi>m</mi> </mrow> </msup> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mi>k</mi> </msup> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mi>m</mi> </msup> <mo>+</mo> <mi>b</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$F(x) \in R_\nu [x]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msub> <mi>R</mi> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$1 \le km < n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>k</mi> <mi>m</mi> <mo><</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>. We establish necessary and sufficient conditions for the integral closedness of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$R_\nu [\theta ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mi>θ</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, expressed explicitly in terms of the coefficients a, b and the integers m, n, k. In particular, when <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\nu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation> is the p-adic valuation on <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathbb {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation>, our results yield criteria to determine the primes dividing <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$[\mathbb {Z}_K: \mathbb {Z}[\theta ]]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mi>K</mi> </msub> <mo>:</mo> <mi mathvariant="double-struck">Z</mi> <mrow> <mo stretchy="false">[</mo> <mi>θ</mi> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$K = \mathbb {Q}(\theta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>=</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathbb {Z}_K$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation> is the ring of integers of K.

Conditions for a Simple Extension of a Valuation Ring to be Integrally Closed

期刊论文
发表日期：

2026年6月16日

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Naveen K. Godara,
Renu Joshi

摘要

Let $\nu $ $ν$ be a Krull valuation of arbitrary rank on a field with valuation ring $R_\nu $ $R_{ν}$ , and let $\theta $ $θ$ be a root of the irreducible polynomial $F(x) = x^{n - km}(x^k + a)^m + b,$ $F (x) = x^{n - k m} {(x^{k} + a)}^{m} + b,$ where $F(x) \in R_\nu [x]$ $F (x) \in R_{ν} [x]$ and $1 \le km < n$ $1 \leq k m < n$ . We establish necessary and sufficient conditions for the integral closedness of $R_\nu [\theta ]$ $R_{ν} [θ]$ , expressed explicitly in terms of the coefficients a, b and the integers m, n, k. In particular, when $\nu $ $ν$ is the p-adic valuation on $\mathbb {Q}$ $Q$ , our results yield criteria to determine the primes dividing $[\mathbb {Z}_K: \mathbb {Z}[\theta ]]$ $[Z_{K} : Z [θ]]$ , where $K = \mathbb {Q}(\theta )$ $K = Q (θ)$ and $\mathbb {Z}_K$ $Z_{K}$ is the ring of integers of K.