Cyclic Galois resolvent ($R(w)={\sum }_{j=0}^{m-1}{\zeta }^{-j}{g}^j(w)$)

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 $\alpha =R(w)\ne 0$ 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.