Valuation ring need not equal the integral closure over a noncomplete field
= Valuation ring need not equal the integral closure over a noncomplete field
Give $\mathbb Q$ its $5$-adic valuation, take $L=\mathbb Q(i)$, and choose the extension associated with the prime $(2+i)$. Then $(2+i)/(2-i)$ lies in the chosen valuation ring but has negative valuation at the conjugate prime. It is therefore not in the integral closure of $\mathbb Z_{(5)}$ in $L$.