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Math Help - GCD

  1. #1
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    GCD

    We have polynomials f(x),g(x) (degree>1), their GCD has degree>1.
    I have to prove that there exist polynomials u(x),v(x) such that:
    f(x).u(x)=g(x).v(x) and deg(u)<deg(g),deg(v)<deg(f). a^-1(x)

    Am I right?
    let d(x) be their GCD, then we have:
    f(x)=a(x).d(x)
    g(x)=b(x).d(x)
    so f(x).a^-1(x)=g(x).b^-1(x) where and b^-1(x) are polynomials u(x),v(x)

    is the proof of existence ok?
    I need help with showing the inequality about degrees. Can anybody help me please?
    Thank you for your help.
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  2. #2
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    Quote Originally Posted by sidi View Post
    We have polynomials f(x),g(x) (degree>1), their GCD has degree>1.
    I have to prove that there exist polynomials u(x),v(x) such that:
    f(x).u(x)=g(x).v(x) and deg(u)<deg(g),deg(v)<deg(f). a^{-1}(x)
    where did a^{-1}(x) come from?? \deg(v) < \deg(f).a^{-1}(x) has no meaning because the RHS of the inequality is not a number!!


    let d(x) be their GCD, then we have:
    f(x)=a(x).d(x)
    g(x)=b(x).d(x)
    so f(x).a^-1(x)=g(x).b^-1(x) where and b^-1(x) are polynomials u(x),v(x)
    the inverse of a polynomial is not necessarily a polynomial. so a^{-1}(x) and b^{-1}(x) might not even exist in your polynomial ring!

    the problem, as you posted, doesn't make any sense and you certainly won't get help if your problem has itself a problem!
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  3. #3
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    Thank you, there was a mistake.
    So the problem is:
    We have polynomials f(x),g(x) (degree>1), their GCD has degree>1.
    I have to prove that there exist polynomials u(x),v(x) such that:
    f(x).u(x)=g(x).v(x) and deg(u)<deg(g),deg(v)<deg(f)

    f(x),g(x) are from F[x], where F is a field

    Thank you for your help.
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  4. #4
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    Quote Originally Posted by sidi View Post
    Thank you, there was a mistake.
    So the problem is:
    We have polynomials f(x),g(x) (degree>1), their GCD has degree>1.
    I have to prove that there exist polynomials u(x),v(x) such that:
    f(x).u(x)=g(x).v(x) and deg(u)<deg(g),deg(v)<deg(f)

    f(x),g(x) are from F[x], where F is a field

    Thank you for your help.
    that's good now! so suppose d(x)=\gcd(f(x),g(x)). we have f(x)=d(x)f_1(x), \ g(x)=d(x)g_1(x). now let u(x)=xg_1(x) and v(x)=xf_1(x). see that

    f(x)u(x)=g(x)v(x). we also have: \deg u(x)= 1 + \deg g_1(x) < \deg d(x) + \deg g_1(x) = \deg (d(x)g_1(x))=\deg g(x). similarly \deg v(x) < \deg f(x).
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