Splitting fields of polynomials

Question

Let $f,g$ $\in$ $F[X]$ non constant polynomials and $K$ a splitting field over $F$. Suppose that $g|f$ in $F[X]$.Prove that $g$ splits to linear factors of $K[X]$.Is $K$ a splitting field of $g$?

Can someone help me with this?

Thank you in advance!

Answer 1

For the first part, let $\alpha$ be a zero of $g$ and $f(x) = g(x)h(x)$, for some polynomial $h$ in $F[x]$. Then we have that $g(\alpha) = 0 \implies f(\alpha) = g(\alpha)h(\alpha) = 0$. So as $\alpha$ is a zero of $f$, we have that $\alpha \in K$, so $g$ splits in $K$.

A simple counterexample to the second question is to consider $f(x) = (x-1)(x^2+1)$ and $g(x) = x-1$ in $\mathbb{Q}$. Obviously the splitting field of $f$ is $\mathbb{Q}(i)$, while the splitting field of $g$ is $\mathbb{Q}$.