Degree of the splitting field

Question

I'm looking for a proof that, if $K/k$ is the splitting field of a polynomial $P$, and $z_1, ..., z_n$ the roots of $P$ in an algebraic closure of $k$, then $$[K : k] | \prod_{i=1}^n [k[z_i]:k]$$ Is it really true ? If not, a proof for $k = \mathbb{Q}$ is sufficient.

Answer 1

This is false. Let $P = x^3 - 2$ (and $k = \mathbb{Q}$). Then the LHS is $6$ and the RHS is $27$.