# Rational Polynomials over a Field

1. Apr 29, 2008

### PingPong

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
None

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. Apr 29, 2008

### Dick

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