Galois Extensions: An Example of Finding a Galois Group

We find the elements of the Galois group of x^4+1 over Q.

[tim-cca]

thank you, nice example for me to understand cyclotomic extension

[antoniocamposrodriguez]

What software do you use for making this kind of videos (handwritten math)?

[guiorasokolovsky]

Please find my mistake. I try to prove that σ(√2)= -√2 ; σ(i)=i
θ(√2) =√2 and θ(i )=-i right from the definitions σ(α)= α^3 and θ(α)=α^5.
From σ(α)= cos (3π/4) + i*sin(3π/4) ⇒ (1/2) cos((π/4)=-(1/2) cos((3π/4) ⇒
σ(√2)= -√2 and i*sin(3π/4) =i*sin(π/4) ⇒ σ(i)=i
So far, so good but:
θ(α)=α^5 = cos (5π/4) + i*sin(5π/4) ⇒ both of θ(√2) θ(i) change signs. Not as claimed in the video. Resulting in
σ(θ(α))=α^7 ⇒ only i changes sign. But still:
σ(θ(α))* σ(θ(α))=θ(α)*θ(α) = σ(α)*σ(α)= Id as claimed.
What is the mistake of my solution ?

[billf]

Its been years since I've studied this stuff, but I'd assume its no coincedence that the powers of alpha that show up in the galois group are precicely those that are coprime with 8?

[spirosgal]

Sorry, aren't the roots of x^2+1, i and -1. Also i dint get why σ squared of sqrt2 is σ(-1)σ(sqrt2) shouldn't it be 2