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Rational Polynomials over a Field

  • Thread starter PingPong
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Homework Statement


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.

Homework Equations


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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:

Answers and Replies

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

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