I was asked to prove that if $a$ is any root of $13x^{4} - 29x^{2} + 13$ then the extension $\mathbb{Q}(a)/\mathbb{Q}$ is a normal extension.
We see that $-a$ is actually another root of the polynomial but I don't know how to show that if $b$ is another root distinct from $a$ and $-a$ then $b\in\mathbb{Q}(a)$.
I know that the polynomial is irreducible since it is a primitive polynomial but doesn't seem to help that is the minimmum polynomial of $a$. Any hint will be appreciate it.