Lemma 15.21.1. Let $R$ be a ring. Let $P(T)$ be a monic polynomial with coefficients in $R$. Let $\alpha \in R$ be such that $P(\alpha ) = 0$. Then $P(T) = (T - \alpha )Q(T)$ for some monic polynomial $Q(T) \in R[T]$.
Proof. By induction on the degree of $P$. If $\deg (P) = 1$, then $P(T) = T - \alpha $ and the result is true. If $\deg (P) > 1$, then we can write $P(T) = (T - \alpha )Q(T) + r$ for some polynomial $Q \in R[T]$ of degree $< \deg (P)$ and some $r \in R$ by long division. By assumption $0 = P(\alpha ) = (\alpha - \alpha )Q(\alpha ) + r = r$ and we conclude that $r = 0$ as desired. $\square$
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