It's not too hard to show that the splitting field of $f(x) = x^3 - 2 \in \mathbb{Q}[x]$ over $\mathbb{Q}$ is $\mathbb{Q}(2^{1/3}, \omega)$ where $\omega$ is a nontrivial third root of unity (a little working shows that it can also be written as $\mathbb{Q}(2^{1/3}, \sqrt{-3})$). However, as one expects of a degree 6 extension, the minimal polynomial is also of degree 6: $x^6 + 9x^4 - 4x^3 + 27x^2 + 36x + 31$. Is there any relation between this higher degree minimal polynomial and the third degree polynomial we started with, considering they both generate the same splitting field? This is meant to be a general question, the above is just an example.
EDIT: A clarification and generalisation of the question. Apologies for the earlier lack of clarity, I see that I made too many assumptions and such.
Given a field $K$ and a polynomial $f(x)\in K[x]$, supposing we can find a field extension $K(\alpha_1, ..., \alpha_n) \cong K[x]/(f)$, is there any relation between $f$ and the minimal polynomial of some linear combination of the $\alpha_i$ (that involves all $\alpha_i$.
The reason I ask this is that I would expect there to be something relating them as both the original polynomial and the resulting minimal polynomial will result in the aforementioned field extension.