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-$$\zeta$$²)(x-$$\zeta$$³)(x-$$\zeta$$⁴).

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.