Let $\sqrt[3]{2}$ be the real cube root of 2.
(a) Find the minimal polynomials of cube root of $2$ and $i$ over $\mathbb{Q}$.
(b) Find the minimal polynomial of $i$ over $\mathbb{Q}(\sqrt[3]{2})$.
(c) Show that $\mathbb{Q}(i\cdot\sqrt[3]{2}) = \mathbb{Q}(i,\sqrt[3]{2})$.
(d) Find the minimal polynomial of $i\cdot\sqrt[3]{2}$ over $\mathbb{Q}$.
I am lost in this setup, and this is not a homework assignment. It is just for something the teacher didn't cover due to lack of time.
For part a:
I got the answer to be x^3 -2 and x^2 + 1, respectively
For part b:
I got the answer to be x^2 + 1
part c: I am unsure what's going on, and need help. can someone explain it please?
part d: answer is x^6 + 4 = 0, right?