This is an offshoot of my other question two days ago. How to apply Hilbert's Irreducibilty theorem?

But it is of independent interest.

Solutions of Inverse Galois Problem for a finite group $G$ give us polynomials whose splitting field has Galois group $G$. I would like to know if there is a way of getting the minimal polynomial (degree equal to order of $G$) for a primitive element of that splitting field.

If this is too cumbersome at least is there a reasonable description of a primitive element?

For example in the case of $S_n$ can we get a recursive description something like this? give two algebraic numbers $\alpha,\beta$ one of them of degree $(n-1)!$ such that $\alpha+\beta$ is of degree $n!$ and generates an $S_n$-Galois extension? (as $S_n$ is generated by an $n$-cycle and an involution, the number of degree $(n-1)!$ above could possibly be fixed by that $n$-cycle).

Known baby example: Considering that cube roots of one and two together generate a Galois extension with $S_3$ as Galois group it is straightforward to see $\omega +\sqrt[3]{2}$ as primitive element and write down its minimal polynomial. (It turns out to be $x^6 + 3x^5 + 6x^4 - 13x^3 - 24x^2 + 33x + 121$, quite a mouthful).