Prove that $x^4-18x^2+6$ is irreducible in $\mathbb{Q}[x]$, and find the degree of its splitting field over $\mathbb{Q}$, and its Galois group. (not a homework problem)
Clearly $x^4-18x^2+6$ is irreducible by Eisenstein with $p=3$. I'm not sure how to find the splitting field and the Galois group. I saw somewhere that for irreducible polynomials, if $\alpha$ is one root, then the other roots are powers of $\alpha$, then the splitting field is simply $\mathbb{Q}(\alpha)$. Is that true?
