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Fundamental Theorem of Algebra. Every non-constant single-variable polynomial with complex coefficients has at least one complex root.

Using the this we can easily show the result with successive divisions, my question is, are there any other ways to prove it?

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    $\begingroup$ This isn't a way to prove it, it is an equivalent statement. $\endgroup$ – Matt Samuel Jan 6 at 12:17
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    $\begingroup$ So if you're asking how to prove that, that is a giant can of worms. Notoriously, there's no known algebraic proof. All of them involve at least some analysis or topology. $\endgroup$ – Matt Samuel Jan 6 at 12:18
  • $\begingroup$ Have a look at Hatchers Algebraic Topology page $31$. $\endgroup$ – drhab Jan 6 at 12:18
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    $\begingroup$ I believe the question is about this result: if every polynomial has at least one root, then every polynomial splits completely . The OP wants to prove it by a method other than successive divisions of polynomials. $\endgroup$ – GEdgar Jan 6 at 13:21
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    $\begingroup$ @Akiva And how is that critical fact proved? Analysis. $\endgroup$ – Matt Samuel Jan 6 at 14:22

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