[FOM] Only one proof

[A. MANI]

On Monday 21 Sep 2009 9:32:44 pm Timothy Y. Chow wrote:
> I think that Melvyn and I were just trying to point out that your use of
> the phrase "the existence of the algebraic numbers" is nonstandard.  
<snip>
> But you meant something like, "the fact that the set of
> algebraic complex numbers is an algebraically closed subfield of C."
> This usage is a little odd, and it was not until just a moment ago that I
> finally figured out that this is what you were trying to say.  Even if we
> formally define "algebraic number" to mean "a complex number that is a
> root of a polynomial equation with rational coefficients," the "existence
> of the algebraic numbers" is a triviality and not an important theorem.

Yes that is very right. All posters speak of 'algebraic numbers' not of 
elements of an extension of a field K being algebraic over K. But even in the 
latter case there is no need of topology. The FT alg is not needed at all for 
"The set of all algebraic numbers over Q forms a subfield of C".  

Best

A. Mani
 


-- 
A. Mani
http://www.logicamani.co.cc