Extending the Fundamental Theorem of Arithmetic to the rationals

[Svein]

I am not sure about the uniqueness, however. Could two different combinations of integers produce the same rational number?

Of course they can $\frac{1}{2}$ and $\frac{17}{34}$ represent the same rational number.

Rationals do not have a unique representation. Every rational number is a member of an equivalence class of combinations of integers.

 

[FactChecker]

Of course they can $\frac{1}{2}$ and $\frac{17}{34}$ represent the same rational number.

Rationals do not have a unique representation.

But they have a unique reduced form to lowest terms, p/q where GCD(p,q)=1. Then both p and q can be factored uniquely into primes where p and q have no common primes.

 

[fresh_42]

But they have a unique reduced form to lowest terms, p/q where GCD(p,q)=1. Then both p and q can be factored uniquely into primes where p and q have no common primes.

The crucial point is the title. The word extension suggests a form of generalization, which it is not. You can't extend from ring to field. No way.

 

[FactChecker]

The crucial point is the title. The word extension suggests a form of generalization, which it is not. You can't extend from ring to field. No way.

There are some formally defined uses of the word "extension", but extending a theorem is not one of those.
In any case, the original question has probably been answered.

 

[Infrared]

The crucial point is the title. The word extension suggests a form of generalization, which it is not. You can't extend from ring to field. No way.

It is a generalization in the sense that it is a correct statement which includes unique factorization in the integers as a special case, not in the sense that it deals with prime elements in $\mathbb{Q}.$

 

[fresh_42]

An application of FTA is something basically different than an extension of FTA.

This is logically relevant as it is algebraically. To throw it all in one pot teaches the wrong motivations. A true statement isn't the same thing as a true classification. All posts above thought to an end would mean: Let's gather all theorems, list them in a book and call it: True. I cannot see how this is doing anyone a favor.