About fundamental theorem of algebra, there is a large collection different demonstrations.

I ask: is there some proof that avoids AC (choice axiom)?

In a general topos (with natural number object) there are the two constructions of real numbers (generalizations of the classical Dedekind and Cauchy constructions) that are different.

In ZF theory, are the Dedekind and Cauchy constructions different? (In the "Cauchy" reals, operates on a real number $r$ through a *choice* of a Cauchy sequence converging to $r$.)