Characterization of Integrally Closed Simple Extensions of Valuation Rings

Abstract

Let $$v$$ be a Krull valuation on a field, with valuation ring $$R_v$$ . Consider the irreducible polynomial $$F(x) = x^n + a(bx^k + c)^m$$ in $$R_v[x]$$ , where $$1 \leq km < n$$ , and let $$\theta$$ be a root of $$F(x)$$ . This paper establishes necessary and sufficient conditions depending solely on the coefficients and exponents $$a, b, c, m, n, k$$ for the ring $$R_v[\theta]$$ to be integrally closed. In the particular case where $$v$$ is the $$p$$ -adic valuation on $$\mathbb{Q}$$ , $$F(x) \in \mathbb{Z}[x]$$ , and $$K = \mathbb{Q}(\theta)$$ , these conditions yield a criterion for determining the primes dividing the index of $$\mathbb{Z}[\theta]$$ in $$\mathbb{Z}_K$$ , the ring of integers of $$K$$ .