# 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.

• Consider $p(x)=x^2-2$. – John Douma Jul 14 '19 at 19:19

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.

• Explain the downvote. – Yves Daoust Jul 14 '19 at 20:04