Prime factorization of rationals

[dodo]

It occurred to me that the rationals Q have also a unique prime factorization, as long as you allow negative exponents on the factorization.

If a/b is a rational, then both a and b have a unique (integer) prime factorization, and the fraction can be expressed uniquely as a product of primes, raised to the difference of the exponents found in the respective prime factors of a and b. Note that a/b does not even need to be reduced for this to work.

I find this a beautiful idea, but I ignore how to use it further, or what else can be constructed using it.

Edit: oops, except for zero... bye-bye to groups, rings, fields...

 

[matt grime]

Good news: this idea is useful. What you've done is come very close to discovering non-archimedian norms, which leads to the study of p-adic anlysis.

Bad news: you can't call it prime factorisation, since everything is a unit.

 

[dodo]

Thanks, Matt. What do you mean, everything is a unit?

 

[CRGreathouse]

Thanks, Matt. What do you mean, everything is a unit?

Evey nonzero rational has a reciprocal that is a nonzero rational. In general, a 'unit' is a member of a ring which has a multiplicative inverse in the ring.

 

[matt grime]

And (unique) prime factorisations, when they exist, are only ever defined 'up to units'. It is just about not mixing and matching your definitions, that's all.