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Lemma 59.81.2. We have

\[ A[s, \frac{1}{\pi s + 1}] = \left(A[t, \frac{1}{\pi ^\ell t + 1}]\right)[s]/(\Phi (s) - t) \]

In particular, the Hopf algebra of $G$ is a monogenic extension of the Hopf algebra of $H$.

Proof. Follows from the discussion above and the shape of $\Phi (s)$. In particular, note that using $\Phi (s) = t$ the element $\frac{1}{\pi ^\ell t + 1}$ becomes the element $\frac{1}{(\pi s + 1)^\ell }$. $\square$

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