A "challenge" that graduate students often do in Algebra for students doing a first course in algebra is: "Prove the Fundamental Theorem of Algebra without using the results of analysis."
To study analysis is necessary axioms that define fields (algebraic axioms) and the supreme axiom (an axiom purely analytical).
It is well known that the supreme axiom implies the Intermediate Value Theorem. Hence it follows Rolle's Theorem, the Mean Value Theorem and the Fundamental Theorem of Calculus, and etc ...
What we can understand about this 'challenge' folkloric we can use all the axioms of fields and equipped with an algebraic construction of the rings of polynomials in one variable to prove the theorem. However we can not use the axiom of the supreme and none of its consequences.
The question that comes to mind is: would it be possible to prove the Fundamental Theorem of Ágebra without using the axiom of supreme?
Always see in books that the answer to this question is no. My question is:
There is a simple ( or intuitive ) explanation that shows why the Fundamental Theorem of Algebra can not be proved without results analysis as described above?
Thank's