arXiv subjects
Publications and source records attributed to L. El Fadil.
Let $K=\mathbb{Q}(α)$ be a number field generated by a complex root $α$ of a monic irreducible polynomial $f(x)=x^{12}-m$, with $m\neq 1$ is a square free rational integer. In this paper, we prove that if $m \equiv 2$ or $3$ (mod 4) and $m\not\equiv \mp 1$ (mod 9), then the number field $K$ is monogenic. If $m \equiv 1$ (mod 8) or $m\equiv \mp 1$ (mod 9), then the number field $K$ is not monogenic.
Let $R$ be a Dedekind ring, $K$ its quotient field, and $L=K(α)$ a finite field extension of $K$ defined by a monic irreducible polynomial $f(x)\in R[x]$. We give an easy version of Dedekind's criterion which computationally improves those versions know in the literature. We further use this result to give a sufficient condition for the integral closedness of $R[α]$ when $f(x)=x^n-a$. In case $R$ is a ring of integers of a number field, we give yet sufficient and necessary conditions for this to hold, generalizing and improving in both cases some known results in this direction. Some highlighting examples are also given.