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The solutions can all be expressed in terms of radicals.

The solutions can all be expressed in terms of radicals.:

R.<x,y,z> = QQ[]
I = R.ideal([x^2+y^2+z^2-2, x^3+y^3+z^3-2, x^4+y^4+z^4-2])
sols = I.variety(QQbar)
for sol in sols:
    print({v: sol[v].radical_expression() for v in sol})

Output:

{z: 0, y: 1, x: 1}
{z: 1, y: 0, x: 1}
{z: 1, y: 1, x: 0}
{z: -2, y: -I*sqrt(2) - 1, x: I*sqrt(2) - 1}
{z: -2, y: I*sqrt(2) - 1, x: -I*sqrt(2) - 1}
{z: -I*sqrt(2) - 1, y: -2, x: I*sqrt(2) - 1}
{z: I*sqrt(2) - 1, y: -2, x: -I*sqrt(2) - 1}
{z: -I*sqrt(2) - 1, y: I*sqrt(2) - 1, x: -2}
{z: I*sqrt(2) - 1, y: -I*sqrt(2) - 1, x: -2}
{z: (1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3) + 1/3/(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3), y: -1/2*(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3)*(I*sqrt(3) + 1) - 1/6*(-I*sqrt(3) + 1)/(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3), x: -1/2*(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3)*(-I*sqrt(3) + 1) - 1/6*(I*sqrt(3) + 1)/(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3)}
{z: (1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3) + 1/3/(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3), y: -1/2*(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3)*(-I*sqrt(3) + 1) - 1/6*(I*sqrt(3) + 1)/(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3), x: -1/2*(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3)*(I*sqrt(3) + 1) - 1/6*(-I*sqrt(3) + 1)/(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3)}
{z: -1/2*(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3)*(I*sqrt(3) + 1) - 1/6*(-I*sqrt(3) + 1)/(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3), y: -1/2*(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3)*(-I*sqrt(3) + 1) - 1/6*(I*sqrt(3) + 1)/(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3), x: (1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3) + 1/3/(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3)}
{z: -1/2*(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3)*(I*sqrt(3) + 1) - 1/6*(-I*sqrt(3) + 1)/(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3), y: (1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3) + 1/3/(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3), x: -1/2*(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3)*(-I*sqrt(3) + 1) - 1/6*(I*sqrt(3) + 1)/(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3)}
{z: -1/2*(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3)*(-I*sqrt(3) + 1) - 1/6*(I*sqrt(3) + 1)/(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3), y: -1/2*(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3)*(I*sqrt(3) + 1) - 1/6*(-I*sqrt(3) + 1)/(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3), x: (1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3) + 1/3/(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3)}
{z: -1/2*(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3)*(-I*sqrt(3) + 1) - 1/6*(I*sqrt(3) + 1)/(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3), y: (1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3) + 1/3/(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3), x: -1/2*(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3)*(I*sqrt(3) + 1) - 1/6*(-I*sqrt(3) + 1)/(1/9*sqrt(3)*sqrt(2) + 1/3)^(1/3)}