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Polynomial Functions : Positive Degree

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The questions are asking for solving h(x) of positive degree.
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1A) Let F be a field and let e(x), f(x), g(x) and h(x) be polynomials in F[x] with h(x) of positive degree. Prove that if e(x) = gcd(g(x),h(x)) and e(x) divides f(x), then there is a polynomial j(x)  F[x] such that g(x)j(x)  f(x) (mod h(x)).

1B) Let F be a field and let d(x), r(x), s(x), and t(x) be polynomials in F[x] with r(x) of positive degree. Prove that if d(x) = gcd(r(x),s(x)) and there is polynomial k(x)  F[x] such that s(x)k(x)  t(x) (mod r(x)), then d(x) divides t(x).
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Solution Summary

Polynomials of positive degree are proven. The proofs of functions are examined.

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1A. Proof:
Since e(x)=gcd(g(x),h(x)), then we can find a(x), b(x) in F[x], such that e(x)=a(x)g(x)+b(x)h(x). Since e(x) divides f(x), then f(x)=r(x)e(x) for some ...

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