Power integral bases in a parametric family of sextic fields (Q2754969)

The author shows the nonexistence of power integral bases in a parametric family of totally complex sextic fields. Let \(n\in {\mathbb N}\) and \(f_n(x)=x^3-nx^2-(n+1)x-1\). (If \(n\geq 3\), then \(f_n(x)\) is totally real.) Let \(\nu=\nu_n\) be a root of \(f_n(x)\) and \(m\) be a square-free positive integer. Consider the two-parametric family \(K={\mathbb Q}(\nu, i\sqrt{m})\) of totally complex sextic fields and the order \({\mathcal O}={\mathbb Z}\left[1,\nu,\nu^2, \omega, \omega\nu, \omega\nu^2\right]\), where \(\omega=(1+i\sqrt{m})/2\;\hbox{if} \;-m\equiv 1 \pmod{4}\), while \(\;\omega=i\sqrt{m} \hbox{if} \;-m\equiv 2, 3 \pmod{4}\).NEWLINENEWLINENEWLINEIt is proven that in the following cases there are no power integral bases in \({\mathcal O}\): NEWLINE\[NEWLINE(i)\;\;n\geq 7 \;\hbox{and} \;m\geq m_0= \begin{cases} 36,& \text{if} \;-m\equiv 1 \pmod{4},\\ 9,& \text{if} -m\equiv 2, 3 \pmod{4},\\ \end{cases} NEWLINE\]NEWLINE NEWLINE\[NEWLINE(ii)\;\;n=3, 4, 5, 6 \;\hbox{and} m\geq 2.NEWLINE\]