[FOM] Improving the Fundamental Theorem of Algebra
joeshipman at aol.com
Wed Nov 16 11:24:44 EST 2005
The "algebraic part" of the Fundamental Theorem of Algebra says that if
a field K has characteristic 0, all odd degree polynomials in K[x] have
roots in K, and all elements of K have square roots in K(i), then K(i)
is algebraically closed.
I can replace "odd degree" with "odd prime degree", and show this is
the best possible.
Is this a new result?
My proof, while tricky, is short and elementary enough that Sylvester
COULD have found it in the 1870's and Artin SHOULD have found it in the
1920's. Although the proof appears "nonconstructive", it is essentially
combinatorial. In particular, it seems likely that, for any given
polynomial of odd degree, the existence of a root is implied (in the
first-order theory of fields) by the existence of roots for an
associated finite set of polynomials of prime degree. Thus,
model-theoretic results on real closed fields are correspondingly
-- Joe Shipman
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