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## Main Question or Discussion Point

This thing splits if we adjoin e^ipi/4.

Let [tex]\zeta[/tex]=e^ipi/4 =[tex]\frac{\sqrt{2}}{2}[/tex]+[tex]\frac{i\sqrt{2}}{2}[/tex]

so x

(x-[tex]\zeta[/tex])(x-[tex]\zeta[/tex]

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

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

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

Since |S

Let [tex]\zeta[/tex]=e^ipi/4 =[tex]\frac{\sqrt{2}}{2}[/tex]+[tex]\frac{i\sqrt{2}}{2}[/tex]

so x

^{4}+1=(x-[tex]\zeta[/tex])(x-[tex]\zeta[/tex]

^{2})(x-[tex]\zeta[/tex]^{3})(x-[tex]\zeta[/tex]^{4}).Then I want to permute these roots so the Galois group is just S

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

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

Since |S

_{4}|=24 something is wrong, but what I can not find what is wrong with the logic.