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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.

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

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

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