On integrally closed simple extensions of valuation rings

Let v be a Krull valuation of a field with valuation ring $R_v$. Let θ be a root of an irreducible trinomial $F(x)=x^n+a{x}^m+b$ belonging to $R_{v}x]$. In this paper, we give necessary and sufficient conditions involving only $a,b,m,n$ for $R_v\theta ]$ to be integrally closed. In the particular case when v is the p-adic valuation of the field $\mathbb{Q}$ of rational numbers, $F(x)\in \mathbb{Z}x]$ and $K=\mathbb{Q}(\theta )$, then it is shown that these conditions lead to the characterization of primes which divide the index of the subgroup $\mathbb{Z}\theta ]$ in $A_K$, where $A_K$ is the ring of algebraic integers of K. As an application, it is deduced that for any algebraic number field K and any quadratic field L not contained in K, we have $A_{KL}=A_K{A}_L$ if and only if the discriminants of K and L are coprime.