ring of integers in a cubic extension of a cyclotomic function field

Question

Let $K=\mathbb{Q}[\omega]$ where $\omega^2+\omega+1=0$ and let $R$ be the polynomial ring $K[x]$. Let $L$ be the field $K(x)[y]$ where $y$ satisfies $y^3=1+x^2$.

Which is the integral closure of $R$ in $L$, why?