91Ó°ÊÓ

The complex numbers \(\mathrm{i} \sqrt{3}\) and \(1+\mathrm{i} \sqrt{3}\) are roots of the quartic \(f=x^{4}-2 x^{3}+7 x^{2}-6 x+12\). Does there exist an automorphism \(\sigma\) of the splitting field extension for \(f\) over \(\mathbb{Q}\) with \(\sigma(\mathrm{i} \sqrt{3})=1+\mathrm{i} \sqrt{3}\) ?

Find a splitting field extension for \(x^{3}-5\) over \(\mathbb{Z}_{7}, \mathbb{Z}_{11}\) and \(\mathbb{Z}_{13} .\)