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Prove that if $f,g$ are coprime polynomials then there exist $a,b$ polynomials such that $af+bg=1$

I think it has something to do with Euclid's algorithm, would like to see a proof.

user11555739
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    If you know how to prove "if f and g are coprime integers then there exist integers a and b such that af+bg=1", then mimic the same proof for polynomials. – user114285 Apr 19 '20 at 17:48
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    You are correct to assume that it has something to do with Euclid's algorithm, since polynomials consitute a euclidean domain, and behaves like integer. The name of the equation you wrote is (generalized) Bézoute's identity for polynomials. Concrete proofs can be found in Kaczorek 2007. Polynomial and Rational Matrices (chapter 1 and 2); Vidyasagar 1985. Control System Synthesis, A Factorization Approach (reprint 2011) (Chapter 4 and Appendices A & B); and Polderman & Willems 1998. Introduction to Mathematical Systems Theory (Appendix B). Hope this helps. – GB. Ha Apr 19 '20 at 18:14

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By Bezout’s identity there exists two polynomials a and b such that $gcd(f,g)=af+bg$. If f and g are coprime then gcd(f,g)=1 and hence there exists polynomials a and b such that $+=1$ (It's exactly the same as the proof for integers: Bezout's identity in $F[x]$).

Alessio K
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