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MONOGENIC EVEN QUARTIC TRINOMIALS

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  13 September 2024

LENNY JONES Open the ORCID record for LENNY JONES [Opens in a new window]
LENNY JONES*
Affiliation:
Department of Mathematics, Shippensburg University, Shippensburg, PA 17257, USA
*
e-mail: doctorlennyjones@gmail.com
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Abstract

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree N is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta )$, where $f(\theta )=0$. We prove that there exist exactly three distinct monogenic trinomials of the form $x^4+bx^2+d$ whose Galois group is the cyclic group of order 4. We also show that the situation is quite different when the Galois group is not cyclic.

Keywords

monogeniccyclicquarticGalois group

MSC classification

Primary: 11R16: Cubic and quartic extensions
Secondary: 11R32: Galois theory

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 111 , Issue 2 , April 2025 , pp. 238 - 243
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

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