The other day, I saw this post about an algebraic proof of Fundamental Theorem of Algebra. In "the most algebraic proof", he uses the intermediate value theorem to prove that every polynomial of odd degree has a real root.
I showed this to my local forum, and then one of the internet strangers opposed that there is an algebraic proof. He says:
Theorem; If the ordered field K has a solution for any odd-order polynomial, then the second-order extension field of K is an algebraically closed field.
and the real number field satisfies the above theorem.
The part that the odd-order equation has a solution is practically equivalent to the intermediate value theorem, but this part does not change much even if it is replaced with the word maximal ordered field.
(Translated by Google Translate so some wording may be weird)
But he didn't provide details and I can't even ask anymore.
The question here is:
What was his method?
Is it really algebraic? (Of course, it depends on what is "algebraic", so just tell me the feeling)
