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Thread: Can any one beat this?(Polynomials)

  1. #1
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    Talking Can any one beat this?(Polynomials)

    If $\displaystyle g.c.d(f(x),g(x))=1$,

    where $\displaystyle f(x) \in \mathbb{K}[x]$ and $\displaystyle g(x) \in \mathbb{K}[x]$,

    where $\displaystyle \mathbb{K}$ is a field of number.

    for $\displaystyle \forall$ $\displaystyle m \in \mathbb{Z}$ and $\displaystyle m>0$

    show that:

    $\displaystyle g.c.d(f(x^m),g(x^m))=1$
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  2. #2
    Senior Member Shanks's Avatar
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    If gcd (f(x), g(x))=1, then there exist u(x) and v(x) in K[x] such that
    u(x)f(x)+v(x)g(x)=1,
    Thus for any integer m>0,
    u(x^m)f(x^m)+v(x^m)g(x^m)=1
    gcd(f(x^m),g(x^m))=1
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