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Keywords:
power integral base; theorem of Ore; prime ideal factorization
power integral base; theorem of Ore; prime ideal factorization
Summary:
Let $K={\mathbb Q}(\alpha)$ be a number field generated by a complex root $\alpha$ of a monic irreducible polynomial $f(x)=x^{18}-m$, $m\neq \mp 1$, is a square free rational integer. We prove that if $ m \equiv 2$ or $3 {\rm(mod }{ 4})$ and $m\not\equiv \mp 1 {\rm(mod }{ 9})$, then the number field $K$ is monogenic. If $ m \equiv 1 {\rm(mod }{ 4})$ or $m\equiv 1 {\rm(mod }{ 9})$, then the number field $K$ is not monogenic.
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