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 $c_0,\ldots,c_{n-1}\in R$ , $$ u^n+c_{n-1}u^{n-1}+\ldots+c_0= $$ 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 $$ a^n+c_{n-1}ba^{n-1}+\ldots+c_0b^n= $$ 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 $u\in R$ .