Lemma 10.136.9. 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 $\beta _ i = -1 \otimes \alpha _ i$.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: