219.sagews

p = x^8 - 12*x^6 + 6*x^4 - 12*x^2 + 1; p

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

xs = p.roots(); xs

[(-sqrt(2*2^(1/4)*sqrt(2*sqrt(2) + 3) + 2*sqrt(2) + 3), 1), (sqrt(2*2^(1/4)*sqrt(2*sqrt(2) + 3) + 2*sqrt(2) + 3), 1), (-sqrt(-2*2^(1/4)*sqrt(2*sqrt(2) + 3) + 2*sqrt(2) + 3), 1), (sqrt(-2*2^(1/4)*sqrt(2*sqrt(2) + 3) + 2*sqrt(2) + 3), 1), (-sqrt(2*2^(1/4)*sqrt(3 - 2*sqrt(2))*I - 2*sqrt(2) + 3), 1), (sqrt(2*2^(1/4)*sqrt(3 - 2*sqrt(2))*I - 2*sqrt(2) + 3), 1), (-sqrt(-2*2^(1/4)*sqrt(3 - 2*sqrt(2))*I - 2*sqrt(2) + 3), 1), (sqrt(-2*2^(1/4)*sqrt(3 - 2*sqrt(2))*I - 2*sqrt(2) + 3), 1)]

[x.abs().exp_simplify() for (x,multiplicity) in xs]

[sqrt(2*2^(1/4)*sqrt(2*sqrt(2) + 3) + 2*sqrt(2) + 3), sqrt(2*2^(1/4)*sqrt(2*sqrt(2) + 3) + 2*sqrt(2) + 3), sqrt(-2*2^(1/4)*sqrt(2*sqrt(2) + 3) + 2*sqrt(2) + 3), sqrt(-2*2^(1/4)*sqrt(2*sqrt(2) + 3) + 2*sqrt(2) + 3), 1, 1, 1, 1]