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Math Help - A field with polynomials and gcd.

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    A field with polynomials and gcd.

    Let F be a field and let f(x),g(x),h(x) and d(x) be polynomials in F[x]. Prove if gcd( f(x),g(x) ) = 1_{F} and f(x) divides g(x)h(x), then f(x)|h(x).
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    Quote Originally Posted by tttcomrader View Post
    Let F be a field and let f(x),g(x),h(x) and d(x) be polynomials in F[x]. Prove if gcd( f(x),g(x) ) = 1_{F} and f(x) divides g(x)h(x), then f(x)|h(x).
    An Euclidean Domain is a Unique Factorization Domain.

    Furthermore we have the property for and field, and F[x].

    That if p(x) is irreducible over F and p(x) divides g(x)h(x) then f(x) divides either g(x) or f(x) divides h(x).

    "Prime" factorize,
    f(x)=r_1(x)r_2(x)...r_k(x)

    Now,
    f(x) divides g(x)h(x).
    Thus,
    r_1(x)...r_k(x) divides g(x)h(x).
    Thus,
    r_1(x) divides g(x)h(x) .... r_k(x) divides g(x)h(x).

    Since r_i(x) are irreducible over F, we have that,

    r_i(x) divides either g(x) or h(x).
    But r_i(x) does not divide g(x) for that will imply that,
    gcd(f(x),g(x))! =1

    Thus,
    r_i(x) divides h(x).
    Hence,
    f(x) divides h(x).
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