[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
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