单向偶四次方三项式

IF 0.6 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society Pub Date : 2024-09-13 DOI:10.1017/s0004972724000510

LENNY JONES

引用次数: 0

摘要

如果 $f(x)$ 在 ${\mathbb Q}$ 上是不可约的，并且 ${1、\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ 是 ${\mathbb Q}(\theta )$ 的整数环的基，其中 $f(\theta )=0$.我们证明恰好存在三个不同的形式为 $x^4+bx^2+d$ 的单元三项式，它们的伽罗瓦群是阶数为 4 的循环群。我们还证明了当伽罗瓦群不是循环群时，情况会截然不同。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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MONOGENIC EVEN QUARTIC TRINOMIALS

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.

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来源期刊

Bulletin of the Australian Mathematical Society 数学-数学

CiteScore

1.20

自引率

14.30%

发文量

149

审稿时长

4-8 weeks

期刊介绍： Bulletin of the Australian Mathematical Society aims at quick publication of original research in all branches of mathematics. Papers are accepted only after peer review but editorial decisions on acceptance or otherwise are taken quickly, normally within a month of receipt of the paper. The Bulletin concentrates on presenting new and interesting results in a clear and attractive way. Published Bi-monthly Published for the Australian Mathematical Society