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

I ask: there is some proof that avoid AC (choise axiom) ?

In a general topos (with natural number object) there are the two construction of real numbers (generalization of the Dedekind and Cauchy classical) 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 choose of a Cauchy sequence converging to $r$).