[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.
Tim
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