Subject: Re: [HM] Euclid and the unique factorization theorem
From: William C Waterhouse (wcw@math.psu.edu)
Date: Fri Feb 11 2000 - 15:19:16 EST
About a week ago, Carlos Cesar de Araujo <carlos.cesar@taskmail.com.br>
wrote a note including the following excerpts from Solomon Bochner's
article "Mathemtical Reflections", Amer. Math. Monthly 1974, 827-852:
(...)
Of course, what had discomfited Dickson was not just the fact that
the fundamental theorem does not occur in Euclid, but, undoubtedly
much more so, that none of the great number-theorists in the 17th
and 18th centuries, like Fermat, Wallis, Euler, Lagrange, etc.,
had chanced upon it either.
(...)
But there is no substance to assertions that the fundamental
theorem had been consciously known to mathematicians before Gauss,
but that they had neglected to make the fact known. We think that
the 17th, and even the 18th, century was not yet ready for the
peculiar kind of mathematical abstraction which the
"fundamental theorem" involves...
It is of course disputable what counts as "consciously known".
It should be made clear, however, that Euler explicitly states the
existence of prime decomposition and presumes its uniqueness.
The first place this occurs is, as one would expect, at the
beginning of the _Elements of Algebra_. I.4 (Art. 41) says
"all composite numbers ... result from the prime numbers
above-mentioned... For, if we find a factor which is not
a prime number, it may always be decomposed and
represented by two or more prime numbers."
Then a bit later (Art. 65), when he discusses divisors of a
number, he says that
"When we have represented any number ... by its simple factors,
it will be very easy to exhibit all the numbers by which it
can be divided."
And of course he does this by taking the primes, then products of
two at a time, then products of three at a time, and so on. Thus
he certainly "recognizes" and uses the uniqueness, though he doesn't
prove it.
The same is true in a more interesting place, the introduction of
the zeta-function in _Introductio in Analysis Infinitorum_ Vol. I,
Chapter 15 (Art. 273). For simplicity he starts without the
exponents that will force convergence, saying:
"If we take all prime numbers and set
P = 1/(1-1/2)(1-1/3)(1-1/5)(1-1/7)(1-1/11)(1-1/13) etc.,
then
P = 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + etc,
where both primes and all numbers arising from primes by
multiplication occur. But since all numbers are either prime
themselves or arise from primes by multiplication, it is clear
that here all whole numbers altogether must be present in
the denominators."
Again he does not state uniqueness but does presume it; that is fairly
clear from the notation and completely clear once he introduces the
exponents and starts doing numerical computation with the series.
William C. Waterhouse
Penn State
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