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Rational Polynomials over a Field

  1. Apr 29, 2008 #1
    1. The problem statement, all variables and given/known data
    Suppose there are two polynomials over a field, f and g, and that gcd(f,g)=1. Consider the rational functions a(x)/f(x) and b(x)/g(x), where deg(a)<deg(f) and deg(b)<deg(g). Show that if a(x)/f(x)=b(x)/g(x) is only true if a(x)=b(x)=0.

    2. Relevant equations

    3. The attempt at a solution
    I've not really gotten any solid ideas here, but these are the few things that have gone through my mind.

    If they are equal then ag=bf, so deg(a)+deg(g)=deg(b)+deg(f). We also have deg(ag)=deg(a)+deg(g)<deg(f)+deg(g) and also deg(bf)=deg(b)+deg(f)<deg(g)+deg(f), so there are no problems there.

    Since f and g are relatively prime, I can write 1=fu+gw for some polynomials u and w. Thus f=(1-gw)/u, and the equality becomes agu=b(1-gw)=b-bgw.

    Am I on the right track? Any hints? Thanks in advance!

    EDIT: Okay, here's something else I've come up with in the past few minutes. Since ag=bf, we have a=bf/g. Thus g divides bf and since, gcd(f,g)=1 so g divides b. But the degree of g is greater than that of b, so b is necessarily the zero polynomial, and a follows similarly.

    Is this okay?
    Last edited: Apr 29, 2008
  2. jcsd
  3. Apr 29, 2008 #2


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    Your last few minutes have been very productive. You convinced me.
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