Assume I have a field $K$ containing all $n$-th roots of unity. You may even assume that $K$ contains an algebraically closed field $\Bbbk$. Assume furthermore that there are $x_1,\ldots,x_k\in K$ and distinct valuations $v_i: K\to \mathbb{Z}\cup\{\infty\}$ with $v_i(x_j)=\delta_{ij}$ (the Kronecker delta). In particular, none of the $x_i$ has a root inside $K$.
Set $L:=K[y_1,\ldots,y_k]$ where $y_i\in\sqrt[n]{x_i}$ is an $n$-th root of $x_i$. Does $L$ have degree $n^k$ over $K$? If yes (which I strongly suspect), how to prove it?
