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| ``Re: Negative primes (seeking comment)''
by mathcam on 2005-10-18 19:26:08 |
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| To a vast majority of people, a prime number refers to the positive version, so I'd suspect that that's the best way to define it in an entry. Number-theorists commonly, by an abuse of terminology, use the term "prime" to semi-interchangeably mean either the prime number itself, or the principal ideal generated by that prime (or even, some times, the collection of prime ideals above a given prime in a field extension).
One drawback to including the negative versions is that unique factorization does not hold, at least not without an additional disclaimer. For example, 6=2*3 and 6=(-2)*(-3) would be two "different" factorizations if we took our list of primes ti include the negative ones. On the plus side, the corrected version of the statement, that we have a unique factorization into *ideals* (since 2 and -2 generate the same ideal) motivates one of the cornerstones of algebraic number theory, i.e. that we *always* have unique factorization of ideals into prime ideals in number fields.
So I'd suggest leaving the definition in terms of positive integers, but perhaps noting that there are benefits obtained by including this more general definition.
Cam
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