On the Monogenity of Quartic Number Fields Defined by <i>x</i><sup>4</sup> + <i>ax</i><sup>2</sup> + <i>b</i>

On the Monogenity of Quartic Number Fields Defined by <i>x</i><sup>4</sup> + <i>ax</i><sup>2</sup> + <i>b</i>

For any quartic number field <i>K</i> generated by a root <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> of an irreducible trin...

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Bibliographic Details
Main Authors:	Lhoussain El Fadil, István Gaál
Format:	Article
Language:	English
Published:	MDPI AG 2025-03-01
Series:	Mathematics
Subjects:	power integral bases theorem of Ore prime ideal factorization common index divisor
Online Access:	https://www.mdpi.com/2227-7390/13/6/905
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Description
Summary:	For any quartic number field <i>K</i> generated by a root <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> of an irreducible trinomial of type <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mo>∈</mo><mi mathvariant="double-struck">Z</mi><mrow><mo>[</mo><mi>x</mi><mo>]</mo></mrow></mrow></semantics></math></inline-formula>, we characterize when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">Z</mi><mo>[</mo><mi>α</mi><mo>]</mo></mrow></semantics></math></inline-formula> is integrally closed. Also for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>=</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>3</mn></mrow></semantics></math></inline-formula>, we explicitly give the highest power of <i>p</i> dividing <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>(</mo><mi>K</mi><mo>)</mo></mrow></semantics></math></inline-formula>, the common index divisor of <i>K</i>. For a wide class of monogenic trinomials of this type, we prove that up to equivalence, there is only one generator of power integral bases in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>=</mo><mi mathvariant="double-struck">Q</mi><mo>(</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We illustrate our statements with a series of examples.
ISSN:	2227-7390