[FOM] Only one proof
pratt at cs.stanford.edu
Fri Sep 18 17:14:13 EDT 2009
...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, you seem to be viewing the elements of splitting fields as numbers.
In my first post I said "algebraic *numbers*", not "roots of
polynomials." Granted this could be clarified, but how did my second
post fail to do so?
Splitting field technology does not prove the FT Alg in the form it has
been understood for at least the last two centuries, namely for
polynomials with real coefficients (to which those f(z) with complex
coefficients easily reduce to via f(z)f(z)^*), since one still has to
coax the splitting fields into C, all known methods for which use analysis.
Nor is it needed, since proofs like the very simple expanding-circle one
I mentioned are complete without splitting field technology.
With this further clarification I stand by my original statement.
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