A question regarding finding the minimal polynomial associated with a field extension . Say we have the field extension $\Bbb Q(w,\sqrt[3]{5})$ over $\Bbb Q$, where w is the primitive cubed root of unity.
I know that the minimum polynomial of $\sqrt[3]{5}$ is $x^3-5$.
I want to figure out the degree of the extension;

I know we can use the tower law $|\Bbb Q(w, \sqrt[3]{5});\Bbb Q|=|\Bbb Q(w, \sqrt[3]{5});\Bbb Q(\sqrt[3]{5})||\Bbb Q(\sqrt[3]{5});\Bbb Q|=x.3$ as $x^3-5$ has degree 3

But how does one find the minimum polynomial of the extension  of degree x
Clearly $w \not \in \Bbb Q(\sqrt[3]{5})$ so the degree is at least $2$. Now notice that $x^2+x+1$ is the minimum polynomial.
Clearly $w \not \in \Bbb Q(\sqrt[3]{5})$ so the degree is at least $2$. Now notice that $x^2+x+1$ is the minimum polynomial.