ACTA issues

## Infinite families of non-monogenic trinomials

Lenny Jones

Acta Sci. Math. (Szeged) 87:1-2(2021), 19-29
213/2021

 Abstract. Let $f(x)\in \mathbb Z [x]$ be monic and irreducible over $\mathbb Q$, with $\deg (f)=n$. Let $K=\mathbb Q (\theta )$, where $f(\theta )=0$, and let $\mathbb Z _K$ denote the ring of integers of $K$. We say $f(x)$ is \emph{non-monogenic} if $\{1,\theta ,\theta^2,\ldots , \theta ^{n-1} \}$ is not a basis for $\mathbb Z_K$. By extending ideas of Ratliff, Rush and Shah, we construct infinite families of non-monogenic trinomials. DOI: 10.14232/actasm-021-463-3 AMS Subject Classification (1991): 11R04; 11R09, 12F05 Keyword(s): monogenic, trinomial, irreducible received 13.2.2021, accepted 2.3.2021. (Registered under 213/2021.)