Lemma 10.136.14. Suppose that $A$ is a ring, and $P(x) = x^ n + b_1 x^{n-1} + \ldots + b_ n \in A[x]$ is a monic polynomial over $A$. Then there exists a syntomic, finite locally free, faithfully flat ring extension $A \subset A'$ such that $P(x) = \prod _{i = 1, \ldots , n} (x - \beta _ i)$ for certain $\beta _ i \in A'$.

**Proof.**
Take $A' = A \otimes _ R S$, where $R$ and $S$ are as in Example 10.136.8, where $R \to A$ maps $a_ i$ to $b_ i$, and let $\beta _ i = -1 \otimes \alpha _ i$. Observe that $R \to S$ is syntomic (Lemma 10.136.13), $R \to S$ is finite by construction, and $R$ is Noetherian (so any finite $R$-module is finitely presented). Hence $S$ is finite locally free as an $R$-module by Lemma 10.78.2. We omit the verification that $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ is surjective, which shows that $S$ is faithfully flat over $R$ (Lemma 10.39.16). These properties are inherited by the base change $A \to A'$; some details omitted.
$\square$

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