Galois Group and Intermediate Field

Question

I just need a detailed explanation of how to go about finding the intermediate fields and galois group of $x^4-x^2-6$. This is not a homework question, I am just confused on how to go about computing these two things. Also, is there a general method of approach when trying to find intermediate fields? Thank you.

Answer 1

Write $f(x)=x^4-x^2-6=(x^2-3)(x^2+2)$ to see that $K=\mathbb Q(\sqrt{3},i\sqrt{2})$ is the splitting field of $f$ over $\mathbb Q$. Then (considering permutations of the roots), $$G(K/\mathbb Q)=\{1,\sigma,\tau,\sigma\tau\}\cong \mathbb Z/2\mathbb Z \times \mathbb Z/2\mathbb Z$$ where $\sigma$ is the $\mathbb Q(i\sqrt 2)$-automorphism given by $\sqrt 3\mapsto -\sqrt 3$, and $\tau$ is the $\mathbb Q(\sqrt 3)$-automorphism given by $i\sqrt 2\mapsto -i\sqrt 2$. So we can see that $\mathbb Q(\sqrt 3)$ and $\mathbb Q(i\sqrt 2)$ are two of the intermediate fields. What is the remaining one?