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Math Help - Rings of Polynomials _ Unique Factorization Theorem

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    Super Member Bernhard's Avatar
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    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  f(x) \in F[X] a non-constant polynomial. Then

    (1)  f(x) = u p_1(x) p_2(x) ... p_s(x) , where  u \in F, u \neq 0 , and each  p_i(x) is a monic irreducible polynomial over F

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

    ================================================== ============================================

    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  f(x) = u p_1(x)p_2(x) ... p_s(x) = vq_1(x)q_2(x) ... q_t(x)

    where u,v  \in F and the  p_i(x) and  q_j(x) are monic and irreducible, then

     p_1(x) | q_1(x)q_2(x) ... q_r(x) ... etc etc


    =========================================

    Can someone show me (formally) why  p_1(x) | q_1(x)q_2(x) ... q_r(x) follows in the argument above

    Peter
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    Re: Rings of Polynomials _ Unique Factorization Theorem

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

    pi(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)....ps(x) = (pi(x))((u/v)[p1(x)...pi-1(x)pi+1(x)...ps(x)])

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

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

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

    then we apply theorem 8.4.5 to show that pi(x) divides some qj(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).
    Last edited by Deveno; October 7th 2012 at 05:17 AM.
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