[FOM] Improving the Fundamental Theorem of Algebra (Update)
joeshipman@aol.com
joeshipman at aol.com
Wed Dec 7 13:36:25 EST 2005
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
More information about the FOM
mailing list