Let F be a field and let e(x), g(x), h(x), and f(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)in F[x] such that

g(x)j(x)=f(x)(modh(x)).

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- Nov 7th 2008, 08:58 AMwvlilgurlmodern alg
Let F be a field and let e(x), g(x), h(x), and f(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)in F[x] such that

g(x)j(x)__=__f(x)(modh(x)). - Nov 7th 2008, 10:42 AMThePerfectHacker
Since $\displaystyle e(x)=\gcd(g(x),h(x))$ it means we can write $\displaystyle e(x) = a(x)g(x)+b(x)h(x)$. But $\displaystyle e(x)|f(x)$ it means $\displaystyle f(x) = e(x)d(x)$. Therefore, $\displaystyle f(x) = e(x)d(x) = a(x)d(x)g(x)+b(x)d(x)h(x)$. It follows that $\displaystyle g(x)j(x) \equiv g(x) ~ \bmod h(x)$ where $\displaystyle j(x)=a(x)d(x)$.