Characterization of Integrally Closed Simple Extensions of Valuation Rings

Let v be a Krull valuation on a field, with valuation ring Rᵥ . Consider the irreducible polynomial F(x) = xⁿ + a(bxᵏ + c)ᵐ in Rᵥ[x] , where 1 ≤ km < n , and let θ 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 Rᵥ[θ] to be integrally closed. In the particular case where v is the p -adic valuation on Q , F(x) ∈ Z[x] , and K = Q(θ) , these conditions yield a criterion for determining the primes dividing the index of Z[θ] in ZK , the ring of integers of K .