Points of $w^2=h_\alpha(t)$ over roots of $h_\alpha$: $w$ is a uniformizer and $h_\alpha'(t)$ is a unit

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Points of $w^2=h_\alpha(t)$ over roots of $h_\alpha$: $w$ is a uniformizer and $h_\alpha'(t)$ is a unit · Prove2Me