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).
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).