[FOM] Only one proof

Vaughan Pratt pratt at cs.stanford.edu
Tue Sep 22 13:39:11 EDT 2009

A. MANI wrote:
> 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".  

Not quite all posters.  This is why I avoided the term "algebraic 
number" in my original reply to Melvyn,

> 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? 

In the kindness of your heart please be so good as to let my original 
problematic phrase "the existence of the algebraic numbers without 
topology" die in peace.  A counterfactual such as "without topology" 
raises the interesting question of which of several previously 
equivalent definitions of "algebraic number" is now the correct one, 
which my reply to Melvyn was worded so as to avoid.

Anyone who considers *that* question sufficiently interesting for FOM 
should start a new thread under a different subject heading.

Vaughan Pratt

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