Something silly question, but I can’t figure out what is wrong.
For example, suppose $K$ be a splitting field of $x^4 - 4x^2 -1$ over $Q$.
Of course, it is irreducible, and zeros are $\pm \sqrt{2+\sqrt{5}}$ and $\pm i \sqrt{\sqrt{5}-2}$. So $K$ would be $Q( \sqrt{2+\sqrt{5}}, i \sqrt{\sqrt{5}-2})$. So the order of galois group would be 8.
However, K is a finite normal extension of Q, which is seperable splitting field over $Q$. Then the order of galois group must be equal to the number of roots of $x^4 - 4x^2 -1$. Then it must be 4…
What did I misunderstood?