On power integral bases for certain pure number fields defined by x 18 - m

Lhoussain El Fadil

Commentationes Mathematicae Universitatis Carolinae (2022)

  • Volume: 62 63, Issue: 1, page 11-19
  • ISSN: 0010-2628

Abstract

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Let K = ℚ ( α ) be a number field generated by a complex root α of a monic irreducible polynomial f ( x ) = x 18 - m , m ≠ ∓ 1 , is a square free rational integer. We prove that if m ≡ 2 or 3 ( mod 4 ) and m ¬ ≡ ∓ 1 ( mod 9 ) , then the number field K is monogenic. If m ≡ 1 ( mod 4 ) or m ≡ 1 ( mod 9 ) , then the number field K is not monogenic.

How to cite

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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 62 63.1 (2022): 11-19. <http://eudml.org/doc/299262>.

@article{ElFadil2022,
abstract = {Let $K=\{\mathbb \{Q\}\}(\alpha )$ be a number field generated by a complex root $\alpha $ of a monic irreducible polynomial $f(x)=x^\{18\}-m$, $m\ne \mp 1$, is a square free rational integer. We prove that if $ m \equiv 2$ or $3 \{\rm (mod \}\{ 4\})$ and $m\lnot \equiv \mp 1 \{\rm (mod \}\{ 9\})$, 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.},
author = {El Fadil, Lhoussain},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {power integral base; theorem of Ore; prime ideal factorization},
language = {eng},
number = {1},
pages = {11-19},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On power integral bases for certain pure number fields defined by $x^\{18\}-m$},
url = {http://eudml.org/doc/299262},
volume = {62 63},
year = {2022},
}

TY - JOUR
AU - El Fadil, Lhoussain
TI - On power integral bases for certain pure number fields defined by $x^{18}-m$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2022
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62 63
IS - 1
SP - 11
EP - 19
AB - Let $K={\mathbb {Q}}(\alpha )$ be a number field generated by a complex root $\alpha $ of a monic irreducible polynomial $f(x)=x^{18}-m$, $m\ne \mp 1$, is a square free rational integer. We prove that if $ m \equiv 2$ or $3 {\rm (mod }{ 4})$ and $m\lnot \equiv \mp 1 {\rm (mod }{ 9})$, 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.
LA - eng
KW - power integral base; theorem of Ore; prime ideal factorization
UR - http://eudml.org/doc/299262
ER -

References

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