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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?

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