On index divisors of certain number fields defined by $x^{11}+ax^2+b$
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In this paper, for any number field $K$ generated by a root $\alpha$ of a monic irreducible trinomial $F(x)=x^{11}+ax^2+b \in \mathbb{Z}[x]$ and for every rational prime $p$, we characterize when $p$ divides the index of $K$. We also describe the prime power decomposition of the index $i(K)$. In such a way we give a partial answer of Problem $22$ of Narkiewicz (\cite{Nar}) for this family of number fields. As an application of our results, if $i(K)\neq1$, then $K$ is not monogenic. We illustrate our results by some computational examples.
Abstract
In this paper, for any number field $K$ generated by a root $\alpha$ of a monic irreducible trinomial $F(x)=x^{11}+ax^2+b \in \mathbb{Z}[x]$ and for every rational prime $p$, we characterize when $p$ divides the index of $K$. We also describe the prime power decomposition of the index $i(K)$. In such a way we give a partial answer of Problem $22$ of Narkiewicz (\cite{Nar}) for this family of number fields. As an application of our results, if $i(K)\neq1$, then $K$ is not monogenic. We illustrate our results by some computational examples.