[FOM] Only one proof

[Rob Arthan]

On Saturday 12 Sep 2009 4:18 am, Vaughan Pratt 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?
>

Once you have that every real polynomial of odd degree has a root, no topology 
is required. The elegant algebraic proof is due to Laplace. See, for example, 
Ebbinghaus et al., Numbers, Springer GTM no 123, appendix to chap. 4.

In his thesis, Gauss excoriated this earlier work as a non-proof. However it 
is only the existence of splitting fields that is controversial. Splitting 
fields are no problem from a modern perspective and are implicit before the 
time of Gauss in the free use of roots of arbitrary equations as symbolic 
quantities that give the right answers in calculations. So Laplace's proof of 
the fundamental theorem of algebra was valid in the 18th century, invalid in 
the 19th and became valid again in the 20th century!

Regards,

Rob.