212.sagews

p = (sqrt(2 - sqrt(2)) + i *sqrt( sqrt(2) - 1 )).minpoly()

p

x^8 - 12*x^6 + 6*x^4 - 12*x^2 + 1

N.<a> = NumberField(p,'y')

a.galois_conjugates?

[a, -a, a^7 - 12*a^5 + 6*a^3 - 12*a, -a^7 + 12*a^5 - 6*a^3 + 12*a]

cs = [ q(sqrt(2 - sqrt(2)) + i *sqrt( sqrt(2) - 1 )).exp_simplify() for q in [x, -x, x^7 - 12*x^5 + 6*x^3 - 12*x, -x^7 + 12*x^5 - 6*x^3 + 12*x] ]

cs

[sqrt(sqrt(2) - 1)*I + sqrt(2 - sqrt(2)), -sqrt(sqrt(2) - 1)*I - sqrt(2 - sqrt(2)), sqrt(sqrt(2) - 1)*I - sqrt(2 - sqrt(2)), sqrt(2 - sqrt(2)) - sqrt(sqrt(2) - 1)*I]

[c.n() for c in cs]

[0.765366864730179 + 0.643594252905583*I, -0.765366864730179 - 0.643594252905583*I, -0.765366864730179 + 0.643594252905583*I, 0.765366864730179 - 0.643594252905583*I]

g = p.galois_group()

g.orbits()

[[1, 5, 2, 6, 3, 7, 8, 4]]