# The galois group of x^4 + 1 over Q

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.