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Is there a proof without using that in $\mathbb C$ every polynomial factorizes into linear ones that every polynomial over $\mathbb R$ factorizes into linear or quadratic ones?

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    $\begingroup$ Given the quadratic formula, this would immediately imply the result for $\mathbb{C}$, would it not? $\endgroup$ – Tobias Kildetoft Nov 9 '16 at 12:03
  • $\begingroup$ @TobiasKildetoft A common way to prove that real polynomials factor into linear and quadratic terms goes via the algebraic closure of $\Bbb C$ and that non-real roots come in complex conjugate pairs. I think the OP want a proof not using $\Bbb C$. $\endgroup$ – Arthur Nov 9 '16 at 12:32
  • $\begingroup$ @Arthur Right, but once you have that proof, the step to $\mathbb{C}$ being algebraically closed is so minor that you would basically have it already. $\endgroup$ – Tobias Kildetoft Nov 9 '16 at 12:34
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    $\begingroup$ @TobiasKildetoft Not quite. You're very close to proving that the $\Bbb C$ contains the roots of any real polynomial, but you still have a little way to go to prove that $\Bbb C$ is algebraically closed. $\endgroup$ – Arthur Nov 9 '16 at 12:47

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