[FOM] Only one proof

[Vaughan Pratt]

Melvyn Nathanson wrote:
>>Pratt wrote:  "...the very existence of the algebraic numbers seems to
>>depend on topology."
> Why?  Algebraic numbers depend only on the definition of a 
> polynomial and a field.  Constructions of roots of polynomials are
> purely algebraic, since at least the 19th century.

Certainly one can, by purely algebraic means, form from the field C
of complex numbers and a polynomial p an extension C' of C containing
a root of p.  But to prove the Fundamental Theorem of Algebra that
way you need the extra step of showing how to collapse C' to C.
How do you do that without appealing to the completeness of C, or
some topological counterpart thereof?

Vaughan Pratt