# Divisibility with GCD [duplicate]

Suppose $$F$$ is a field, let $$f(x), g(x), h(x) \in F[x]$$ and $$d(x)=\gcd(f(x),g(x))$$. If $$f(x) \mid h(x)$$ and $$g(x) \mid h(x)$$, prove that $$f(x)g(x) \mid d(x)h(x)$$.

• What have you tried? Where are you stuck? – Robert Shore Oct 9 '19 at 15:43
• $f,g\mid h\iff {\rm lcm}(f,g)\mid h,\,$ and $\,{\rm lcm}(f,g) = fg/\gcd(f,g).\,$ Most of the proofs for integers in the linked dupes also work here since $F[x]$ is a PID so enjoys GCDs, unique factorization, Bezout identity etc. – Bill Dubuque Oct 9 '19 at 15:50

$$F[X]$$ is a principal domain, so $$d(x)=a(x)f(x)+b(x)g(x)$$
Write $$h(x)=u(x)f(x)=v(x)g(x)$$.
$$d(x)h(x)=$$
$$(a(x)f(x)+b(x)g(x))h(x)=$$
$$a(x)v(x)f(x)g(x)+b(x)u(x)f(x)g(x)$$