[FOM] The theory of fields; professional puzzlement

joeshipman@aol.com joeshipman at aol.com
Mon Jan 16 02:37:50 EST 2006

Recent work I have done in elementary algebra has some interesting 
foundational implications for the theory of fields, and I'd like to 
solicit comment on it from algebraists, proof theorists and model 
theorists on the FOM list. I have some more general observations at the 
end which I encourage everyone to comment on!

In what follows, [n] denoted the statement "every polynomial of degree 
n has a root", which is a sentence in the first-order theory of fields.

Let AF denote the conjunction of the standard axioms for fields.

Let (n) abbreviate the sum of n 1's in the language of fields; thus (5) 
abbreviates 1+(1+(1+(1+1)))).

Theorem (Van der Waerden, following E. Artin, inspired by one of 
Gauss's proofs of the Fundamental Theorem of Algebra):
If a field of characteristic 0 satisfies [n] for n=2 and all odd n, 
then it is algebraically closed.

Theorem 1:
A field is algebraically closed iff [p] holds for each prime p. If any 
prime is omitted there is a counterexample. The following 
axiomatizations are "perfect" (no axiom may be omitted):

For general algebraically closed fields: {AF, [2], [3], [5], [7], 
For algebraically closed fields of characteristic p: {(p)=0, AF, [2], 
[3], [5], [7], [11], ...}
For algebraically closed fields of characteristic 0: {AF, ~((2)=0), 
[2], ~((3)=0), [3], ~((5)=0), [5], ... }

Theorem 2:
If {p1,p2,...} is an infinite set of primes, and a field K satisfies 
each [p_i], then either K is algebraically closed, or there exists a 
prime p not in {p1,p2,...} such that [n] is true in K iff n is not a 
multiple of p.

Theorem 3:
The sentence ([d1]&[d2]&...&[d_m]) --> [n] is true in all fields iff 
the following condition holds:
for every finite group G acting without fixed points on {1,2,...,n}, 
the additive semigroup generated by the indexes of proper subgroups of 
G contains one of the d's.

What I find interesting is that even though a great deal of interesting 
work has been done in the model theory of fields, the striking 
elementary facts detailed above were unknown.  In particular,

1) nobody seems to have even asked if the axiomatizations for 
algebraically closed fields were the best possible

2) nobody seems to have discovered that the Artin/v.d.Waerden version 
of the Fundamental Theorem of Algebra could be extended to 
characteristic p, let alone that "2 or odd" could be replaced by "prime"

3) nobody (except, apparently, John H. Conway) suspected the existence 
of the rich theory of finitary implications between "degree axioms" 
that Theorem 3 establishes, even though every one of those 
implications, when true, is a first-order consequence of the standard 
axioms for fields and could have been discovered by anyone in the last 
two centuries.

The unsettling moral I draw from this (because I know that what I did 
was not particularly diificult) is that this was a "hole" in 
mathematics so classical that every math Ph.D. has encountered it, and 
that there may be many more such undiscovered treasures.

Perhaps mathematicians don't try hard enough to discover new things in 
familiar territory, because if they spend a few years to reach a 
"frontier" they are not only much more likely to discover new things, 
but there will be far fewer people capable of reaching that particular 
frontier and competing with them.

Am I being too cynical? (I hope so, because that would mean what I did 
was cleverer than I'd thought it was!)

-- Joseph Shipman

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