[FOM] Improving the Fundamental Theorem of Algebra (Update)

[joeshipman@aol.com]

Here is a precise statement of my results so far. If anyone knows of 
any anticipations of these results, please send me a reference. 
 
The "Fundamental Theorem of Algebra" is often taken to be a statement 
about the particular fields of real and complex numbers, but it has an 
"algebraic part" that applies to all fields. One version of this is 
stated as follows: 
 
(1) If a field K of characteristic 0 has roots for all polynomials 
whose degree is 2 or an odd number, then K is algebraically closed. 
 
Another standard version, which is an easy corollary of the above, is 
 
(2) If K is a real field where all positive elements have square roots 
and all odd degree polynomials have roots, then K(i) is algebraically 
closed. 
 
I can now extend these results in several ways. 
 
(3) "Odd degree" may be replaced by "odd prime degree" above. 
 
(4) Result (3) is also valid for fields of characteristic p. (And (1) 
is also valid for fields of characteristic p of course; this may 
already be a new result but I doubt it.) 
 
(5) These results are best possible in the sense that for each prime p, 
there are fields such that all polynomials of degree not divisible by p 
have roots, but which are not algebraically closed. (I don't think this 
fact is new, I'm just quoting it to show that the earlier results are 
optimal). 
 
(6) If the set of degrees for which all polynomials have roots is 
infinite, then either K is algebraically closed or there is exactly one 
"bad prime" p and all polynomials whose degree is not divisible by p 
have roots. 
 
Now, define {a1,a2,...,a_n} --> b to mean "if all polynomials whose 
degree is a1 or a2 or ... or a_n have roots, then all polynomials of 
degree b have roots". (Each one of these statements is expressible in 
the first-order language of fields.) 
 
(7) If {a1,a2,...,a_n} --> b, then one of the a's is >= b, except for 
the special case {2,3}-->4. 
 
(8) If {a1,a2,...,a_n} --> b, then every prime dividing b divides one 
of the a's. 
 
(9) If b divides a, then {a} --> b (trivial) 
 
(10) If c is the binomial coefficient (n choose k) = n!/(k!(n-k)!), 
then {k,c} --> n in fields of characteristic 0, and {k,c,p} --> n in 
fields of characteristic p. (This is the difficult one, that does most 
of the work in the proofs of (3) and (4)). 
 
(11) If all polynomials of degree a1+a2 have roots, then either all 
polynomials of degree a1 have roots, or all polynomials of degree a2 
have roots. We can write this as {a1+a2} --> (a1 or a2). 
 
(12) If the primes dividing b are q1,q2,...,q_n, then for all 
sufficiently large k, {q1,q2,...,q_n,k} --> b in fields of 
characteristic 0, and {p,q1,q2,...,q_n,k} --> b in fields of 
characteristic p. 
 
I am currently investigating whether there are any finitary relations 
{a1,...,a_n} --> b that are not formal consequences of (7)-(12). By 
model-theoretic arguments, any such relation that is true for 
characteristic 0 must be true for all but finitely many characteristics 
p. 
 
-- Joseph Shipman