Polynomials Over a Field

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silvermane
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Homework Statement


Take two polynomials f(x) and g(x) over a field, and suppose that gcd(f,g)=1.
Show that any rational function, [tex]\frac{h(x)}{f(x)g(x)}[/tex] can be written in the form
[tex](\frac{p(x)}{f(x)})[/tex] + [tex](\frac{q(x)}{g(x)})[/tex] for some polynomials, p and q.



The Attempt at a Solution


I claim that any rational function [tex](\frac{h(x)}{f(x)g(x)})[/tex] can be written in the form [tex](\frac{p(x)}{f(x)})[/tex] + [tex](\frac{q(x)}{g(x)})[/tex] for some polynomials, p and q.

We're given that the gcd(f,g) = 1, so we could write them as a linear combination of their gcd:
1 = u(x)f(x) + w(x)g(x).
We divide both sides by f(x) and g(x) to obtain:
[tex](\frac{1}{f(x)g(x)})[/tex] = [tex](\frac{u(x)}{g(x)})[/tex] + [tex](\frac{w(x)}{f(x)})[/tex]
We could then multiply both sides by h(x) to obtain:
[tex](\frac{h(x)}{f(x)g(x)})[/tex] = [tex](\frac{h(x)*u(x)}{g(x)})[/tex] + [tex](\frac{h(x)*w(x)}{f(x)})[/tex]

Now, let p(x) = h(x)*u(x) and q(x) = h(x)*w(x). Thus, we have that
[tex](\frac{h(x)}{f(x)g(x)})[/tex] = [tex](\frac{p(x)}{g(x)})[/tex] + [tex](\frac{p(x)}{f(x)})[/tex]
and we've proven our claim.

I just want to make sure that what I've done here is correct. Just so that I understand the material and what's going on. Thank you for your help in advance! :blushing:
 

Answers and Replies

  • #2
22,089
3,293
seems fine.
It's a cool proof actually...
 
  • #3
silvermane
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:blushing:
Thanks! I had a lot of fun writing it once I figured the whole thing out.
Thanks for checking it for me!
 

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