Lemma 15.21.2. Let $R$ be a ring. Let $P(T)$ be a monic polynomial with coefficients in $R$. There exists a finite free ring map $R \to R'$ such that $P(T) = (T - \alpha )Q(T)$ for some $\alpha \in R'$ and some monic polynomial $Q(T) \in R'[T]$.

Proof. Write $P(T) = T^ d + a_1T^{d - 1} + \ldots + a_0$. Set $R' = R[x]/(x^ d + a_1x^{d - 1} + \ldots + a_0)$. Set $\alpha$ equal to the congruence class of $x$. Then it is clear that $P(\alpha ) = 0$. Thus we win by Lemma 15.21.1. $\square$

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