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.
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: