Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f(x)=x^{n}+ax^{3}+bx+c$</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> <mi>n</mi> </msup> <mo>+</mo> <mi>a</mi> <msup> <mi>x</mi> <mn>3</mn> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> </mrow> </math></EquationSource> </InlineEquation> be the minimal polynomial of an algebraic integer <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation> over the rationals with certain conditions on a,  b,  c,  and n. Let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$K={\mathbb Q}(\theta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>=</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a number field and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {O}_{K}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">O</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation> be the ring of integers of K. In this article, we characterize all the prime divisors of the discriminant of f(x) which do not divide the index of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\mathbb Z}[\theta ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">[</mo> <mi>θ</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {O}_{K}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">O</mi> <mi>K</mi> </msub> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> As an interesting corollary, we establish necessary and sufficient conditions for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${\mathbb Z}[\theta ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">[</mo> <mi>θ</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> to be integrally closed. Finally, we investigate the types of solutions to certain differential equations associated with the polynomial f(x).

On characterization of Monogenic number fields associated with certain quadrinomials and its applications

期刊论文
发表日期：

2025年12月5日

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Tapas Chatterjee,
Karishan Kumar

摘要

Let $f(x)=x^{n}+ax^{3}+bx+c$ $f (x) = x^{n} + a x^{3} + b x + c$ be the minimal polynomial of an algebraic integer $\theta $ $θ$ over the rationals with certain conditions on a, b, c, and n. Let $K={\mathbb Q}(\theta )$ $K = Q (θ)$ be a number field and $\mathcal {O}_{K}$ $O_{K}$ be the ring of integers of K. In this article, we characterize all the prime divisors of the discriminant of f(x) which do not divide the index of ${\mathbb Z}[\theta ]$ $Z [θ]$ in $\mathcal {O}_{K}.$ $O_{K} .$ As an interesting corollary, we establish necessary and sufficient conditions for ${\mathbb Z}[\theta ]$ $Z [θ]$ to be integrally closed. Finally, we investigate the types of solutions to certain differential equations associated with the polynomial f(x).