proof of finite extensions of Dedekind domains are Dedekind

Let $R$ be a Dedekind domain with field of fractions $K$. If $L/K$ is a finite extension of fields and $A$ is the integral closure of $R$ in $L$, then we show that $A$ is also a Dedekind domain.

We procede by splitting the proof up into the separable and purely inseparable cases. Letting $F$ consist of all elements of $L$ which are separable over $K$, then $F/K$ is a separable extension and $L/F$ is a purely inseparable extension.

First, the integral closure $B$ of $R$ in $F$ is a Dedekind domain (see proof of finite separable extensions of Dedekind domains are Dedekind). Then, as $A$ is integrally closed and contains $B$, it is equal to the integral closure of $B$ in $L$ and, therefore, is a Dedekind domain (see proof of finite inseparable extensions of Dedekind domains are Dedekind).