Suppose $L/K$ is a finite Galois extension of fields of degree $n$. Suppose that we know an irreducible polynomial $f\in K[x]$ such that $L\cong K[x]/(f)$. Suppose also that we know the Galois group of the extension as a subgroup of $S_{[L:K]}$ (meaning that we know its permutation representation on the roots $\alpha_1,\ldots,\alpha_n$ of $f$).
Now suppose we are given a polynomial $g\in K[x_1,\ldots,x_n]$. My question is: is there any generic algorithm to compute the minimal polynomial of $g(\alpha_1,\ldots,\alpha_n)\in L$? By "generic" I mean something that does not make use of the nature of the fields we are looking at (for example I'm not interested in looking at the specific case where $K,L$ are number fields). Actually in my specific case $K$ and $L$ have characteristic $2$.
