Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Colloquium Mathematicum / Online First articles

Abstract

For any number field $K$ generated by a root $\alpha$ of a monic irreducible trinomial $F(x)=x^{10}+ax^m+b \in \mathbb Z[x]$ with $1\leq m\leq 9$ and for every rational prime $p$, we give sufficient conditions which guarantee that $p$ divides the index of $K$. We also calculate $\nu _p(i(K))$ in each case. For $m=1$, we show that the index of $K$ is either $1$ or a power of $3$ for any $(a,b)\in \mathbb Z^2$, and we characterize when $3$ divides $i(K)$. As an application, we show that if $i(K)\neq 1$, then $K$ is not monogenic. We illustrate our results by some computational examples.