Algebraic numbers are the roots of polynomial equations with integer coefficients.
A polynomial equation with integer coefficients is a*x0 + b*x1 + c*x2 + d*x3 + e*x4 + f*x5+ . . . = 0 where a, b, c, d, e, f . . . are integers.
x is algebraic if and only if it is the solution to such an equation.
(NOTE: The exponent of 1, and the whole term of x0, would normally be left out, as their presence does not alter the result. They were included only to make the formula clearer. -Monzo)