Finding degree of extension of splitting field of a polynomial

Let $$K$$ be the splitting field of $$f(x)=x^{3}+πx+6$$ over $$F =\mathbb{Q}(π)$$ and $$K'$$ be the splitting field of $$g(x)=x^{3}+ex+6$$ over $$F'=\mathbb{Q}(e)$$.

Is $$[K:F]=[K':F']$$ ?

$$f(x)$$ is irreducible over $$\mathbb{Q}(π)$$ and it has only one real root and two complex roots. So $$[K:F]=6$$ (as degree of extension of splitting field divides $$n!$$, where $$n$$ is the degree of the polynomial). Similarly $$[K':F']=6$$. Is this correct?