The galois group of x^4 + 1 over Q

[Daveyboy]

This thing splits if we adjoin e^ipi/4.
Let \zeta=e^ipi/4 =\frac{\sqrt{2}}{2}+\frac{i\sqrt{2}}{2}
so x⁴+1=

(x-\zeta)(x-\zeta²)(x-\zeta³)(x-\zeta⁴).

Then I want to permute these roots so the Galois group is just S₄.

But, Q(\zeta)=Q(i,\sqrt{2}) and [Q(i,\sqrt{2}):Q]=4 (degree)

I have the theorem that Galois group \leq degree of splitting field over base field.

Since |S₄|=24 something is wrong, but what I can not find what is wrong with the logic.

 

[matt grime]

Then I want to permute these roots so the Galois group is just S₄.
...but what I can not find what is wrong with the logic.

What's wrong is that you've forgotten the definition of a Galois group - it is the group of field automorphisms, not just the arbitrary permutation of roots.