[FOM] Improving the Fundamental Theorem of Algebra

joeshipman@aol.com joeshipman at aol.com
Sun Nov 27 01:58:48 EST 2005

Ever since Gauss, it has been known that if every odd degree polynomial 
in a field K has a root, and every element of K has a square root in 
K(i), then K(i) is algebraically closed.

I recently proved that "odd degree" may be weakened to "odd prime 
degree" here, and that this is the best possible (for any prime p, 
there are fields in which all polynomials with degree not divisible by 
p have roots, but some of degree p don't).

This proof has been reviewed by other mathematicians and I am writing 
it up for publication.

I'm not claiming that this is a very deep result; the proof is only 2 
pages long and quite elementary, depending only on very basic facts 
about symmetric polynomials, binomial coefficients, and algebraic 

What I'm interested in discussing on the FOM list is this:

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?

If this theorem was never proved before, why not? It's an obvious 
question to ask whether the hypothesis "Odd degree polynomials have 
roots" can be weakened, especially since the theorem for odd degrees 
has been regarded as the "algebraic part" of the Fundamental Theorem of 
Algebra, which generalizes to all fields of characteristic 0. (The 
"analytic" or "topological" part has been, at a minimum, that real 
polynomials of odd degree have roots in the real numbers; if you don't 
push the algebra as far, you need more analysis/topology, such as a 
winding number argument or an application of Liouville's theorem).

I'll be happy to email the proof to anyone who wishes to see it.

-- Joseph Shipman

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