Let $F$ a Galois over $K$, and let $B$ be a subfield of $F$ such that : $K \le B\le F$ $\Rightarrow$ $F$ a Galois over $B$

PROOF:
$F$ is a splitting field of $f \in K[x]$ separable over $K$.
$F=K(a\_1,...,a\_n)$ and $a\_i$ are the roots of $f$ in $F$.
Then $F=B(a\_1,...,a\_n)$ and $F$ is a splitting field of $f$ over $F$.
I have to show that $f$ is separable over B, how?

Can somebody help me with this proof?

EDIT:

From my notes: "If $p$ is an irreducible factor of $f$ in $K$ then $p$ is product of irreducible factors over $B$" why?