Explanation of Fundamental Theorem of Algebra 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.
The author possibly means that the theorem cannot be proven by mere algebra, because rational coefficients can generate irrational roots, which require more powerful methods, such as limits.
The author possibly means that the theorem cannot be proven by mere algebra, because rational coefficients can generate irrational roots, which require more powerful methods, such as limits.