Suppose that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f(x)=x^4+Ax^3+Bx^2+Ax+1\in \mathbb {Z}[x]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>x</mi> <mn>4</mn> </msup> <mo>+</mo> <mi>A</mi> <msup> <mi>x</mi> <mn>3</mn> </msup> <mo>+</mo> <mi>B</mi> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> <mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We say that f(x) is monogenic if f(x) is irreducible over <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\mathbb {Q}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\{1,\theta ,\theta ^2,\theta ^3\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <msup> <mi>θ</mi> <mn>2</mn> </msup> <mo>,</mo> <msup> <mi>θ</mi> <mn>3</mn> </msup> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> is a basis for the ring of integers of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\mathbb {Q}}(\theta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$f(\theta )=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. For each possible Galois group G that can occur in the two cases of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$A\ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$B=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$AB\ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>B</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we determine all monogenic polynomials f(x) with Galois group G.

Monogenic Reciprocal Quartic Polynomials and their Galois Groups

期刊论文
发表日期：

2026年3月26日

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Lenny Jones

摘要

Suppose that $f(x)=x^4+Ax^3+Bx^2+Ax+1\in \mathbb {Z}[x]$ $f (x) = x^{4} + A x^{3} + B x^{2} + A x + 1 \in Z [x]$ . We say that f(x) is monogenic if f(x) is irreducible over ${\mathbb {Q}}$ $Q$ and $\{1,\theta ,\theta ^2,\theta ^3\}$ ${1, θ, θ^{2}, θ^{3}}$ is a basis for the ring of integers of ${\mathbb {Q}}(\theta )$ $Q (θ)$ , where $f(\theta )=0$ $f (θ) = 0$ . For each possible Galois group G that can occur in the two cases of $A\ne 0$ $A \neq 0$ with $B=0$ $B = 0$ , and $AB\ne 0$ $A B \neq 0$ , we determine all monogenic polynomials f(x) with Galois group G.