Splitting fields of $(X^3-2)(X^2-2)$

Question

For the polynomial $(X^3-2)(X^2-2)\in\mathbb Q$ we have its splitting field $L=\mathbb Q(\sqrt[3]2,\sqrt2)$. How can the degree of $[L:\mathbb Q]$ be obtained?

What would the splitting field be in $\mathbb F_3$, e.g. of $(X^3+1)(X^2+1)$?

Answer 1

Hint:

First you need to adjoint a primitive root of unity $\xi \neq 1$ such that $\xi^3 = 1$. You may take $\xi = \cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3}$. The sppliting field $L = \mathbb{Q}[\sqrt[3]{2}, \sqrt{2}, \xi]$. Now notice that

$$[\mathbb{Q}[\sqrt[3]{2}]:\mathbb{Q}] = 3\\ [\mathbb{Q}[\sqrt{2}]:\mathbb{Q}] = 2\\ [\mathbb{Q}[\xi]:\mathbb{Q}] = 2$$

To see the latter notice that $\xi$ is root of $q(x) = X^2+X+1 \in \mathbb{Q[X]}$ and $q(x)$ is irreducible over $\mathbb{Q}$.