# UFD’s are integrally closed

Theorem: Every UFD is integrally closed.

Proof: Let $R$ be a UFD, $K$ its field of fractions, $u\in K,u$ integral over $R$. Then for some $c_{0},\ldots,c_{n-1}\in R$,

 $u^{n}+c_{n-1}u^{n-1}+\ldots+c_{0}=0$

Write $u=\frac{a}{b},a,b\in R$, where $a,b$ have no non-unit common divisor (which we can assume since $R$ is a UFD). Multiply the above equation by $b^{n}$ to get

 $a^{n}+c_{n-1}ba^{n-1}+\ldots+c_{0}b^{n}=0$

Let $d$ be an irreducible divisor of $b$. Then $d$ is prime since $R$ is a UFD. Now, $d\lvert a^{n}$ since it divides all the other terms and thus (since $d$ is prime) $d\lvert a$. But $a,b$ have no non-unit common divisors, so $d$ is a unit. Thus $b$ is a unit and hence $u\in R$.

Title UFD’s are integrally closed UFDsAreIntegrallyClosed 2013-03-22 15:49:25 2013-03-22 15:49:25 rm50 (10146) rm50 (10146) 6 rm50 (10146) Theorem msc 13G05