Lemma 10.120.11. A unique factorization domain is normal.

**Proof.**
Let $R$ be a UFD. Let $x$ be an element of the fraction field of $R$ which is integral over $R$. Say $x^ d - a_1 x^{d - 1} - \ldots - a_ d = 0$ with $a_ i \in R$. We can write $x = u p_1^{e_1} \ldots p_ r^{e_ r}$ with $u$ a unit, $e_ i \in \mathbf{Z}$, and $p_1, \ldots , p_ r$ irreducible elements which are not associates. To prove the lemma we have to show $e_ i \geq 0$. If not, say $e_1 < 0$, then for $N \gg 0$ we get

which contradicts uniqueness of factorization in $R$. $\square$

## 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Your email address will not be published.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: