[FOM] Only one proof

[Vaughan Pratt]

joeshipman at aol.com wrote:
> That leaves Feit/Thompson as my best example of an important theorem 
> with essentially only one proof. Can anyone think of another? (It's not 
> enough that there is one idea common to all proofs unless that idea is 
> the only difficulty -- I'm looking for a theorem where all proofs are 
> essentially similar both globally and locally.) 

As we know from Gauss there are strikingly different proofs of the 
Fundamental Theorem of Algebra (that every univariate polynomial with 
complex coefficients has a complex root), although they all rely on 
topology.  (One would imagine there should also be a discrete 
nontopological proof using algebraic numbers, the obstacle there seems 
to be that the very existence of the algebraic numbers seems to depend 
on topology.)

But what about the Fundamental Theorem of Arithmetic?  All proofs I'm 
aware of seem to boil down to one version or another of Euclid's GCD 
algorithm.  Is there a different proof?

Vaughan Pratt