# [FOM] Perfect Powers

A. Mani a_mani_sc_gs at yahoo.co.in
Tue Dec 28 17:49:25 EST 2010

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