[FOM] New Proof of Fundamental Theorem of Arithmetic
joeshipman@aol.com
joeshipman at aol.com
Fri Sep 25 16:18:05 EDT 2009
Yes, Rotman himself calls it a new proof. He writes:
***
Here is a new proof of the fundamental theorem of arithmetic.
Corollary 5.53. Every integer n ≥ 2 has a factorization into primes,
and the prime
factors are uniquely determined by n.
Proof. Since the group In is finite, it has a composition series; let
S1, . . . , St be the factor
groups. Now an abelian group is simple if and only if it is of prime
order, by Proposition
2.107; since n = |In| is the product of the orders of the factor groups
(see Exercise 5.36
on page 287), we have proved that n is a product of primes. Moreover,
the Jordan–H¨older
theorem gives the uniqueness of the (prime) orders of the factor
groups. •
***
so he gets the credit for using Jordan-Holder to prove the F T of
Arithmetic.
But maybe my later simplification, which proves Euclid's Lemma and
avoids Jordan-Holder, is new?
-- JS
-----Original Message-----
From: George McNulty <mcnulty at mailbox.sc.edu>
See page 282 of Joseph Rotman's Advanced Modern
Algebra (a thousand page graduate algebra mass) for a
version of the proof using the Jordan-H\"older Theorem
to get the uniqueness of the prime factorization of integers.
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