Let F be a field and let f(x),g(x),h(x) and d(x) be polynomials in F[x]. Prove if gcd( f(x),g(x) ) = 1_{F} and f(x) divides g(x)h(x), then f(x)|h(x).

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- Mar 27th 2007, 06:02 PMtttcomraderA field with polynomials and gcd.
Let F be a field and let f(x),g(x),h(x) and d(x) be polynomials in F[x]. Prove if gcd( f(x),g(x) ) = 1_{F} and f(x) divides g(x)h(x), then f(x)|h(x).

- Mar 27th 2007, 08:04 PMThePerfectHacker
An Euclidean Domain is a Unique Factorization Domain.

Furthermore we have the property for and field, and F[x].

That if p(x) is irreducible over F and p(x) divides g(x)h(x) then f(x) divides either g(x)**or**f(x) divides h(x).

"Prime" factorize,

f(x)=r_1(x)r_2(x)...r_k(x)

Now,

f(x) divides g(x)h(x).

Thus,

r_1(x)...r_k(x) divides g(x)h(x).

Thus,

r_1(x) divides g(x)h(x) .... r_k(x) divides g(x)h(x).

Since r_i(x) are irreducible over F, we have that,

r_i(x) divides either g(x) or h(x).

But r_i(x) does not divide g(x) for that will imply that,

gcd(f(x),g(x))! =1

Thus,

r_i(x) divides h(x).

Hence,

f(x) divides h(x).