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Math Help - modern alg

  1. #1
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    modern alg

    Let F be a field and let e(x), g(x), h(x), and f(x) be polynomials in F[x] with h(x) of positive degree. Prove that if e(x) = gcd(g(x),h(x)) and e(x) divides f(x), then there is a polynomial j(x)in F[x] such that
    g(x)j(x)= f(x)(modh(x)).
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  2. #2
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    Quote Originally Posted by wvlilgurl View Post
    Let F be a field and let e(x), g(x), h(x), and f(x) be polynomials in F[x] with h(x) of positive degree. Prove that if e(x) = gcd(g(x),h(x)) and e(x) divides f(x), then there is a polynomial j(x)in F[x] such that
    g(x)j(x)= f(x)(modh(x)).
    Since e(x)=\gcd(g(x),h(x)) it means we can write e(x) = a(x)g(x)+b(x)h(x). But e(x)|f(x) it means f(x) = e(x)d(x). Therefore, f(x) = e(x)d(x) = a(x)d(x)g(x)+b(x)d(x)h(x). It follows that g(x)j(x) \equiv g(x) ~ \bmod h(x) where j(x)=a(x)d(x).
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