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Re: CVS commit: src/games/factor
On Tue, May 18, 2010 at 10:46:52AM -0700, Paul Goyette wrote:
> Neither side has offered any authoritative sources for their
> respective arguments. Since neither of the primary participants
> appears to be a recognized authority on the subject, the entire
> discussion is nothing more than "amusing".
Please pick a random book in a math library and check the definition.
I haven't bothered to name specific ones because I am only aware of the
two definitions I gave. I know quite a lot of topics in math that don't
have that much agreement, but I am simply not aware of anyone recognized
authority to disagree with this. Too many results depend on the set of
primes not including 1. But to answer the question that has been raised
by the "forward thinking" crowd, this is the reason why the definitions
exist as they are. Many of the current theorems involving primes either
don't work for unit elements or are completely useless and obvious. In
other words, they either wouldn't add understanding or would have to
exclude the unit elements. An example for the former is the factor ring
Z/pZ. With p==1, you get the special 0-ring, which is extremely boring
as it contains only one element and all operations are constant. In that
way it can even be harmful in some cases, as you can't pick 1 != 0.
A more involved example of the latter is the earlier mentioned theorem
about prime factorisation. The unique prime factorisation is a very
powerful concept and hard to justify to give up. Note that when talking
about Z, the uniqueness is down to equivalence of primes, e.g. it
doesn't guarantee that you can't pick different signs for the individual
primes, but it says that it is the only difference. The number of prime
factors doesn't change and the absolute value doesn't either.
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