On sextic integral bases using relative quadratic extention

Abstract

Let $K=\mathbb{Q}(\theta)$ be a cubic number filed and $P(X)=X^3-aX-b$ ($a,b$ in $\ZZ$), the monic irreducible polynomial of $\theta$. In this paper we give a sufficient conditions on $a$,$b$ which ensure that $\theta$ is a power basis generator, also we give conditions on relative quadratic extension to be monogenic. As a consequence of this theoretical result we can reach an integral basis of some sextic fields which Neither algebraically split nor arithmetically split.