Two relatively prime polynomials

Question

show that two polynomials f,g in K[x] are relatively prime if and only if 1 is in I(f,g). I(f,g) being the linear combinations of the two polynomials. I dont know how to start this since shouldnt 1 have to be in there in order for them to be relatively prime?

What is K? If $K=\Bbb Z/4\Bbb Z$,then for $f=2$ and $g=x$ the assertion is not true. — viplov_jain, Feb 18 '14 at 22:02

score 2 · Answer 1 · answered Feb 18 '14 at 22:05

2

This result is called Bézout's identity for polynomials and it's proof is based on Euclidean algorithm.

answered Feb 18 '14 at 22:05

For that $K[X]$ must be an Euclidean Domain. right? – viplov_jain Feb 18 '14 at 22:06
Yes since we should do an euclidean division. – Feb 18 '14 at 22:08
So, what are the minimum conditions on K so that $K[X]$ is an ED. – viplov_jain Feb 18 '14 at 22:09

Two relatively prime polynomials

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