# Question about a proof of FTA in A classical Introduction to modern number theory

I just started to work on A classical introduction to modern number theory by K. Ireland and M. Rosen and I have a question.

In the first chapter, they prove the following lemma:

Every nonzero integer can be written as a product of primes.

Then they mention the following.

By collecting terms we can write $n=p_1^{\alpha_1}\cdots p_m^{\alpha_m}$, where the $p_i$'s are primes and the $a_i$'s are nonnegative integers. [...] The exponents $a(p)$ are nonnegative integers and of course $a(p)=0$ for all but finitely many primes.

Then after proving some other lemmas, he proves that $a(p)=\text{ord}_pn$ (where $\text{ord}_pn$ is the largest nonnegative integer $t$ for which $p^t\mid n$).

My question is: isn't he (either Rosen or Ireland) assuming already (implicitly of course) that $a(p)=\text{ord}_pn$ when he says that we can collect terms and write $n=p_1^{\alpha_1}\cdots p_m^{\alpha_m}$? (or he's just proving that $\text{ord}_pn$ is unique?)

I understand that given a prime $p,$ $\text{ord}_p$ is a function that maps integers to nonnegative integers (and $+\infty$ if the input is $0$), and that this implies that the prime factorization of a nonzero integer $n$ is unique (up to order), but only if we "collect terms" as the authors do, but how do we exactly define the $a(p)$'s?

When we prove (the "usual" proof that everyone knows) the uniqueness part of the FTA by induction on $n$ (in this book they also prove the existence by induction), we don't need to "collect terms", so, we do not really assume anything "illegal".

I hope that my question is clear and I really appreciate any help (and I'm sorry if this is a silly question, maybe I'm just a bit confused).

• Given any representation of $n$ as a product of primes, we can certainly collect equal primes and give the representation in exponential form. That says nothing yet about the essential uniqueness of the representation. Sep 5, 2015 at 4:32
• Ok, I get it now. I was confused...thanks!
– CIJ
Sep 5, 2015 at 4:39
• You are welcome. Sep 5, 2015 at 5:02

Given any representation of $n$ as a product of primes, we can certainly collect equal primes and produce a representation in "exponential" form. That says nothing about the essential uniqueness of the representation, it is just a book-keeping move, since we are taking commutativity of multiplication for granted.