Integral closure of $\mathbb C[t]$ in $\mathbb C(t)[W]/(W^2-h_\alpha)$: exactly the elements $a(t)+b(t)\,w$

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Integral closure of $\mathbb C[t]$ in $\mathbb C(t)[W]/(W^2-h_\alpha)$: exactly the elements $a(t)+b(t)\,w$ · Prove2Me