[FOM] Perfect Powers

[A. Mani]

Let Q be the ordered field of rationals with usual operations (+, ., -, -1, 
\vee, \wedge, 0, 1) (\vee, \wedge  are used for the total order to make it an 
algebra).

For p, q \in N, the well known result: 

p^{1/q} is rational iff p is a perfect qth power
 
is typically proved via the prime factorization theorem.

But it can also be proved by much simpler direct methods provided an 
additional 'non-algebraic' function F: Q -> N is permitted. E.g floor or 
ceiling function (these are not term or polynomial functions w.r.t the basic 
operations assumed). 

Can the latter proof be made algebraic?


Best

A. Mani




-- 
A. Mani
ASL, CLC, AMS, CMS
http://www.logicamani.co.cc