I need to solve the following system of equations:
(a^2 - a + 1) = b
-5 + 7 a = b^2
16 - 9 a - 12 a^2 = b^3

I tried:

Solve[{a^2 - a + 1 == b,  7 a - 5 == b^2, -12 a^2 - 9 a + 16 == b^3}, {b, b^2, b^3}]

But got an error, b^2 is not a valid variable

The answer is b^3 +3b^2+12b-13 = 0

Note: since b = a^2 - a +1, b^2 = 7a - 5, b^3 = -12a^2 - 9a + 16, then b^3 +12 b = -21a + 28 and b^3 + 3b^2 + 12b = -13. This monic polynomial is irreducible by Eisenstein Criterion. Thus, it's the minimal polynomial.

This problem arose from the following: If P(x) = x^3 +3x -3 is a polynomial with a as a root, find the minimal polynomial of b=a^2 - a + 1.

Thanks