Geodesic
On power integral bases for certain pure number fields defined by $x^{18}-m$
Commentationes Mathematicae Universitatis Carolinae, Tome 63 (2022) no. 1, pp. 11-19 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.14712/1213-7243.2022.005
Classification : 11R04, 11R16, 11R21
Keywords: power integral base; theorem of Ore; prime ideal factorization 
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El Fadil, Lhoussain. On power integral bases for certain pure number fields defined by $x^{18}-m$. Commentationes Mathematicae Universitatis Carolinae, Tome 63 (2022) no. 1, pp. 11-19. doi: 10.14712/1213-7243.2022.005

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