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 $c_0,\mathrm{\dots },c_{n-1}\in R$,

$$u^n+c_{n-1}{u}^{n-1}+\mathrm{\dots }+c_0=0$$

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

$$a^n+c_{n-1}b{a}^{n-1}+\mathrm{\dots }+c_0{b}^n=0$$

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