As John Conway said (see sibling comment, or e.g., [1]):
> Every nonzero rational number has a unique factorization into powers of distinct primes.
As you note, (-1)^2 = 1. But if you read carefully, you'll see that the factorization of -100/3 is uniquely:
-1 * 4 * 1/3 * 5 * 1 ...
whereas the factorization of 100/3 is uniquely:
1 * 4 * 1/3 * 5 * 1 ...
Where it's true that the representation using primes with exponents is not unique, it is true that the representation using powers of primes is unique. That is, your issue regarding (-1)^2 is that there are infinitely many representations of 1 or -1 having the form (-1)^x, but if you evaluate (-1)^x (x being integral, of course) you'll only get one of two numbers.
And yes, changing the definition of "prime" does change some special cases -- it removes some (such as extending unique factorization to negative rationals), and adds others (wherever primes are assumed positive).
> Every nonzero rational number has a unique factorization into powers of distinct primes.
As you note, (-1)^2 = 1. But if you read carefully, you'll see that the factorization of -100/3 is uniquely:
whereas the factorization of 100/3 is uniquely: Where it's true that the representation using primes with exponents is not unique, it is true that the representation using powers of primes is unique. That is, your issue regarding (-1)^2 is that there are infinitely many representations of 1 or -1 having the form (-1)^x, but if you evaluate (-1)^x (x being integral, of course) you'll only get one of two numbers.And yes, changing the definition of "prime" does change some special cases -- it removes some (such as extending unique factorization to negative rationals), and adds others (wherever primes are assumed positive).
[1] http://swc-alpha.math.arizona.edu/video/2009/2009ConwayLectu... (mention around 7:00)