Thread: Rings of Polynomials _ Unique Factorization Theorem

I am reading Papantonopoulou - Algebra - Ch 8 Rings of Polymonials

On page 251 (see attachment) Papantonopoulou states the Unique Factorization Theorem as follows:

Let F be a field and a non-constant polynomial. Then

(1) , where , and each is a monic irreducible polynomial over F

(2) Except for the order of the irreducible facotrs, the factorization in (1) is unique

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The proof of (1) is fine, but I am having trouble with the start of the uniqueness proof which reads as follows: (see attachment)


Use induction on deg f(x) = n.

If n=1, uniqueness holds.

So now assume uniqueness holds for all polynomials of degree < n = deg f(x).

If

where u,v and the and are monic and irreducible, then

... etc etc


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Can someone show me (formally) why follows in the argument above

Peter

the key fact is that p_i(x) is irreducible. in particular, p_i(x) is NOT a unit, and so deg(p_i(x)) > 0.

(irreducible elements are non-units by definition: we want to eliminate "trivial factorizations" like u = (u²)(u^-1), or even worse: u = (-1)(-1)(u)).

p_i(x) clearly divides the "whole product" (that is, f(x)), and what we really want to show is that it divides "the q-part".

so consider (1/v)(f(x)). we have:

(1/v)f(x) = (u/v)p₁(x)....p_s(x) = (p_i(x))((u/v)[p₁(x)...p_i-1(x)p_i+1(x)...p_s(x)])

which shows (1/v)f(x) is a product of p_i(x) and some other polynomial in F[x] (namely: (u/v)[p₁(x)...p_i-1(x)p_i+1(x)...p_s(x)]).

but: (1/v)f(x) = (1/v)(vq₁(x).....q_r(x)) = (v/v)(q₁(x)...q_r(x)) = q₁(x)...q_r(x),

so p_i(x) must divide this polynomial as well.

then we apply theorem 8.4.5 to show that p_i(x) divides some q_j(x) (theorem 8.4.5 is KEY; it proves "irreducibles are prime" in F[x]).

this shows that F[x] is a unique factorization domain....actually we didn't even need "the polynomial-ness" (if that's a word) of F[x], the ONLY thing we need is the "euclidean function" d (deg(f) in this case), so that we have a division algorithm to find the gcd of 2 elements of F[x]: ALL euclidean domains are unique factorization domains, the proof is pretty much the same, and in fact, is the SAME proof used to show integers have a unique (up to sign, and order of prime factors) factorization into prime numbers (which are precisely the "irreducibles" of the ring Z).

note we used the fact that F is a field in one crucial step...asserting that 1/v exists. it is important that neither u nor v be 0 (the units of F[x], though, ARE the units of F, that is: the non-zero elements of F).