Conditions for a Simple Extension of a Valuation Ring to be Integrally Closed

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 $$F(x) = x^{n - km}(x^k + a)^m + b,$$ where $$F(x) \in R_\nu [x]$$ and $$1 \le km < n$$ . 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, k. In particular, when $$\nu $$ is the p-adic valuation on $$\mathbb {Q}$$ , our results yield criteria to determine the primes dividing $$[\mathbb {Z}_K: \mathbb {Z}[\theta ]]$$ , where $$K = \mathbb {Q}(\theta )$$ and $$\mathbb {Z}_K$$ is the ring of integers of K.