Lemma 10.123.1. Let $\varphi : R \to S$ be a ring map. Suppose $t \in S$ satisfies the relation $\varphi (a_0) + \varphi (a_1)t + \ldots + \varphi (a_ n) t^ n = 0$. Then $\varphi (a_ n)t$ is integral over $R$.

**Proof.**
Namely, multiply the equation $\varphi (a_0) + \varphi (a_1)t + \ldots + \varphi (a_ n) t^ n = 0$ with $\varphi (a_ n)^{n-1}$ and write it as $\varphi (a_0 a_ n^{n-1}) + \varphi (a_1 a_ n^{n-2}) (\varphi (a_ n)t) + \ldots + (\varphi (a_ n) t)^ n = 0$.
$\square$

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