Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

Abstract

A monic polynomial $f(x)\in \mathbb Z[x]$ of degree $n$ that is irreducible over $\mathbb Q$ is called cyclic if the Galois group over $\mathbb Q$ of $f(x)$ is the cyclic group of order $n$, while $f(x)$ is called monogenic if $\{1,\theta,\theta^2,\ldots , \theta^{n-1}\}$ is a basis for the ring of integers of $\mathbb Q(\theta )$, where $f(\theta )=0$. In this article, we show that there do not exist any monogenic cyclic trinomials of the form $f(x)=x^4+cx+d$. This result, combined with previous work, proves that the only monogenic cyclic quartic trinomials are $x^4-4x^2+2$, $x^4+4x^2+2$ and $x^4-5x^2+5$.