[FOM] Improving the Fundamental Theorem of Algebra

Timothy Y. Chow tchow at alum.mit.edu
Sun Nov 27 16:26:55 EST 2005

Joe Shipman wrote:
> If this theorem was ever proved before, why do ALL textbooks on algebra
> require all odd degrees in their treatment of real closed fields, and
> not mention, even as an exercise or in a footnote, that this assumption
> is stronger than necessary?

Getting into textbooks is such a haphazard process that surprise should be 
reserved for the rare cases in which the textbook treatment is optimal, 
not for the typical case in which it is not.

> If this theorem was never proved before, why not?

Do we expect classical results to be optimal?  My personal impression is 
that optimization is a relatively uncommon phenomenon.  It seems to occur 
when some result or theory captures someone's aesthetic fancy, or when 
there is some particular application driving the investigation that 
requires our utmost efforts.  It seems that the latter doesn't apply in 
this case, so we are left with the former.  Both the classical version and 
Shipman's improvement are attractive enough that I am somewhat---but not 
excessively---surprised that this result is new.  Aesthetic taste is a 
fickle thing.


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