On primes dividing the index of a quadrinomial

Abstract

Let ℤK denote the ring of algebraic integers of an algebraic number field K=ℚ(𝜃) where the algebraic integer 𝜃 is a root of an irreducible quadrinomial f(x)=xn+axn−1+bxn−2+c belonging to ℤ[x] with a2=4b. We give necessary and sufficient conditions involving only a,b,c,n for a prime p to divide the index of the subgroup ℤ[𝜃] in ℤK. As a consequence, we obtain necessary and sufficient conditions for ℤK to be equal to ℤ[𝜃]. Moreover, when ℤK≠ℤ[𝜃], we provide an explicit formula for the index [ℤK:ℤ[𝜃]] in some cases.