# Calculating degree of a finite Kummer Extension

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?

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By Kummer theory, it's enough to show that the subgroup of $K^*/K^{*n}$ generated by the $x_i$ is isomorphic to $(\mathbb{Z}/n)^k$. The tuple of valuations, reduced mod $n$, give such an isomorphism. – Kevin Sep 19 '11 at 19:37
Can you name a reference? I think I just don't happen to know a good textbook that covers Kummer theory. – Jesko Hüttenhain Sep 19 '11 at 21:25
Lang's Algebra, Chapter VI, Theorem 8.1. – Kevin Sep 20 '11 at 2:16