[FOM] Conway on generalizing F.T. of A.

joeshipman@aol.com joeshipman at aol.com
Fri Dec 23 19:33:09 EST 2005

Let A_n be the axiom "all polynomials of degree n have roots", for an 
arbitrary field K. While generalizing the Fundamental Theorem of 
Algebra, I had established various finite implications between these 
axioms, mainly following from (A_k & A_d) --> A_n whenever d = (n 
choose k) = n!/(k!(n-k!)).

(In characteristic p you may additionally need to assume A_p).

But I thought there might be deeper examples, based on the kind of 
combinatorics of permutation groups that John Conway explored in his 
work on finite versions of the Axiom of Choice in 1970. I went to see 
Conway today, and we were able to establish, for example, A30 --> A8, 
and (A3 & A10) --> A6, which go beyond what my binomial coefficient 
techniques could show.

A related phenomenon: if A6 is true (all sextic equations have roots), 
then not only can you get A2 (trivially), A3 (trivially), and A4 
(because quartics can be reduced to quadratics and cubics), but any 
quintic which does NOT have roots must have a cyclic Galois group 
(because of combinatorics involving transitive representations of S5 in 

It looks like there is a fruitful area for research here. It's curious 
no one seems to have examined these finitary relationships!

-- JS

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