$\newcommand\Q{\mathbb Q}$Suppose I have an irreducible polynomial $f\in\Q[x]$ and suppose I know the roots say $r_1,\dots,r_n\in \bar \Q$. I want to know if there is an easy way to "compute" the smallest number field $K$ such that we have $$a_0(r_1,\dots, r_n) + a_1(r_1,\dots, r_n)x + \dots + a_k(r_1,\dots, r_n)x^k \in K[x]$$ where $a_0,\dots, a_k \in \Q[x_1,\dots, x_n]$. I am actually only interested in knowing $[K:\Q]$.
I suppose this is "strongly" dependent on how $a_0,\dots, a_k$ are constructed. In my particular case, I want to prove that $[K:\Q]$ is actually always even. I guess I can go into details of the $a_i's$ but I am not sure if this will help in solving $[K:\Q]$. For me, the multivariate polynomials $a_0,\dots,a_k$ are not symmetric but they are certainly not random (i.e. there are non-trivial permutations of the $n$-indeterminates that preserve the $a_i$'s). I do not know if any of these information helps.
Even a specific example that could give me a start on how to solve this kind of problem is very much appreciated.