From Axler's Linear Algebra Done Right:
Fundamental Theorem of Algebra:
Every nonconstant polynomial with complex coefficients has a zero.
He proves this using complex anaylsis (which I havn't learnt yet). But then he notes:
All proofs of the Fundamental Theorem of Algebra need to use some analysis, because the result is not true if C is replaced, for example, with the set of numbers of the form $c+ di$ where $c, d$ are rational numbers.
Does this mean that the theorem isn't true when we restrict our polynomials to having rational complex coefficients?
edit That's how I read it, axler clearly meant the theorem doesn't hold if the roots were restricted to rational complex numbers.