= Cyclic Galois resolvent
{title2=$R(w)=\sum_{j=0}^{m-1}\zeta^{-j}g^j(w)$}
For a cyclic <Galois extension> $M/K$ of degree $m$, choose a generator $g$ and a primitive $m$th root of unity $\zeta\in K$. Reindexing the displayed sum gives $gR(w)=\zeta R(w)$. <Linear independence of distinct field embeddings> ensures this sum is not identically zero as a function of $w$. Choosing $\alpha=R(w)\ne0$ therefore gives $m$ distinct conjugates $\zeta^j\alpha$, while $\alpha^m\in K$. Consequently $M=K(\alpha)$ is the splitting field of $X^m-\alpha^m$. Independence follows by taking a shortest supposed linear relation among distinct automorphisms, evaluating at $uv$ and subtracting a suitable multiple of the relation at $v$ to remove one term but not all terms.
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