[FOM] Improving the Fundamental Theorem of Algebra

[joeshipman@aol.com]

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 
improved.

-- Joe Shipman