On the index of generalized trinomials over Valued fields

Let $\nu$ be a Krull valuation of arbitrary rank on a field with valuation ring $R_{\nu }$, and let $\theta$ be a root of the irreducible polynomial $F(x)=(x^k+c{)}^m-a{x}^n\in R_{\nu }[x]$ and $1\le n<km$. We establish necessary and sufficient conditions for the integral closedness of $R_{\nu }[\theta ]$, expressed explicitly in terms of the coefficients $a$, $b$, and the integers $m$, $n$, and $k$. In particular, when $\nu$ is the $p$-adic valuation on $\mathbb{Q}$, our results yield criteria for determining the primes dividing the index $[{\mathbb{Z}}_K:\mathbb{Z}[\theta ]],$ where $K=\mathbb{Q}(\theta )$ and ${\mathbb{Z}}_K$ denotes the ring of integers of $K$.