Question

let F be field that is contained in a large field K.let f(x)and g(x) be two polynomial in F[x]that are coprime in F[x].show that f(x)and g(x)are also coprime in K[x]

Answer 1

Well, I will preface this response by saying I am not an expert in this particular topic. However, I did find several resources which made some sense about it. I'll leave it here for your or others' evaluation.
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f(x) and g(x) are two coprime polynomials in F[x].
We should first establish what it means of these two polynomials to be coprime. A similar concept exists for numbers, namely if you have gcd(a,b) = 1, or in other words ta linear combination of a and b equal to 1, then a and b are coprime. There is likely a similar property akin to polynomials you could sketch.

Showing the grander scheme with K[x] seems to me a simple process of comparison. F[x] would be contained in K[x], so K[x] should be able to support similar concepts perhaps such as those of coprime-ness.
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Anyways, good luck!

Answer 2

thank u