Normal closure of a field extension adjoined by a $3$-rd root of unity

Question

As stated in the Wikipedia article, the normal extension of the field $\mathbb Q(\sqrt[3]{2})$ is $\mathbb Q(\sqrt[3]{2},e^{i\frac{\pi}{3}})$. Does there exist a polynomial $P$ of degree $5$, $w$ is a root of $P$, such that the normal extension of $\mathbb Q(w)$ is $\mathbb Q(w,e^{i\frac{\pi}{3}})$?

Answer 1

Let $a=\sqrt[3]{2}$, let $b=\exp\bigl(\frac{2i\pi}{3}\bigr)$, and let $w=a+b$.

Then using Maple's Groebner basis package, we get

• $P(w)=0$, where $P\in\mathbb{Q}[x]$ is irreducible and given by $P(x)=x^6+3x^5+6x^4+3x^3+9x+9$.$\\[6pt]$
• $a=\bigl({\large{\frac{1}{9}}}\bigr) \bigl( 2w^5+3w^4+6w^3-6w^2+9w+18 \bigr) $.$\\[6pt]$
• $b=\bigl({\large{\frac{1}{9}}}\bigr) \bigl( -2w^5-3w^4-6w^3+6w^2-18 \bigr)$.

It follows that $\mathbb{Q}(w)=\mathbb{Q}(a,b)$.