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Math Help - great common divisor

  1. #1
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    great common divisor

    Prove: let F be a field and let g(x) and h(x) be polynomials in F[x] with gcd(g(x),h(x))=1. Let a, b be elements of F with ab. Then gcd(h(x)-ag(x), h(x)-bg(x))=1.
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  2. #2
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    Quote Originally Posted by apple2009 View Post
    Prove: let F be a field and let g(x) and h(x) be polynomials in F[x] with \text{gcd}(g(x),h(x))=1. Let a, b be elements of F with a\neq b. Then \text{gcd}(h(x)-ag(x), h(x)-bg(x))=1.
    Let p(x) be an irreducible monic divisor of h(x)-ag(x) and h(x)-bg(x). Then p(x) divides bh(x)-abg(x) and ah(x)-abg(x), and therefore also [ah(x)-abg(x)]-[bh(x)-abg(x)]=(a-b)h(x) and h(x). This in turn means p(x) is a factor of g(x). Since 0\leq\text{deg }p(x)\leq\text{deg gcd}(h(x),g(x))=0, then p(x)=1, which implies the conclusion.
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