On integrally closed simple extensions of valuation rings

Abstract

Let v be a Krull valuation of a field with valuation ring Rv. Let θ be a root of an irreducible trinomial F(x)=xn+axm+b belonging to Rv[x]. In this paper, we give necessary and sufficient conditions involving only a,b,m,n for Rv[θ] to be integrally closed. In the particular case when v is the p-adic valuation of the field Q of rational numbers, F(x)∈Z[x] and K=Q(θ), then it is shown that these conditions lead to the characterization of primes which divide the index of the subgroup Z[θ] in AK, where AK 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 AKL=AKAL if and only if the discriminants of K and L are coprime.