Find the ordered pair (a,b) of real numbers for which x^2+ax+b has a non-real root whose cube is 1.

If x^3 = 1, then x^3 - 1 = 0.

Factorizing gives (x - 1)(x^2 + x + 1) = 0.

If the root x is non-real and its cube is 1, then x^2 + x + 1 = 0.

Comparing coefficients gives a = 1, b = 1.