[FOM] Only one proof

Timothy Y. Chow tchow at alum.mit.edu
Tue Sep 15 18:28:12 EDT 2009

Vaughan Pratt <pratt at cs.stanford.edu> wrote:
>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, you seem to be conflating the two statements

1. the field of complex numbers is algebraically closed; and
2. there exists an algebraic closure of the rationals.


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