this is just gcd of two polynomials. to prove unique suppose not then arrive at a contradiction. this is only an existence and uniqueness proof
Can you help me with this problem:
I have to prove that if gcd(f(x),g(x))=1 then there exist unique u(x) and v(x) such that u(x)f(x)+v(x)g(x)=1 and deg u(x)< deg g(x) , deg v(x)< deg f(x).
This problem seems so strange. How to prove this?
Existence is given by euclidean algorithm: Here it is: An important consequence of the Euclidean algorithm is finding integers and such that
This can be done by starting with the equation for , substituting for from the previous equation, and working upward through the equations. Now just prove uniqueness. Then you are done. i.e. euclidean algorithm is valid for any ring of polynomials.
those who replied to your question forgot that we also need to have and the problem is not trivial! anyway, i'll assume that the polynomials are over
a field. we know that there exist polynomials and such that now there exist polynomials and such that where either
or also there exist polynomials and such that where either or
so: which is possible only if why? therefore: and the existence part is done. for the uniqueness
suppose we have with and then we will have:
thus, since we must have which is possible only if because
hence and therefore: which gives us: