I am trying to understand how Liouville's Theorem would apply to complex numbers.
alpha an irrational algebraic, then there exist a constant, c, depending on alpha such that for all rationals, the inequality is satisfied.
c/q^d <= abs(alpha-(p/q))
Assume x is a complex irrational algebraic. In the theorem's inequality, would alpha be seperately the Re(x) and Im(x) of the complex number or would it be the magnitude of x?
If x is complex algebraic, is its real and imaginary parts necessarily algebraic?
I am asking because I'm trying to prove Liouville's theorem for alpha a complex number.