[ripsdmzo] Equality of algebraic numbers

Given two algebraic numbers, determine if they are equal; equivalently, given an algebraic number, determine if it is equal to zero.

Intuitively, this problem seems difficult, perhaps intractable.  However, if the algebraic number is specified as a polynomial with integer coefficients, it does not seem so hard: is the constant term zero?  Perhaps the difficulty is in specifying which root of the polynomial.