Monogenity of composition of quadrinomials

Abstract An irreducible monic polynomial f ( x ) ∈ Z [ x ] $f(x)\in \mathbb{Z}[x]$ is said to be monogenic if Z [ θ ] $\mathbb{Z}[\theta ]$ is the ring of integers of the number field Q ( θ ) $\mathbb{Q}(\theta )$ for a root θ of f ( x ). Let f ( x ) = x n + ax n −1 + bx n −2 + c , a 2 = 4 b , and g ( x ) ∈ Z [ x ] $g(x)\in \mathbb{Z}[x]$ be such that f ( g ( x )) is irreducible over Q $\mathbb{Q}$ . In this article, we establish a criterion that characterises the primes dividing the index, [ Q ( α ) : Z [ α ] ] $[\mathbb{Q}(\alpha ):\mathbb{Z}[\alpha ]]$ where α is a root of the composition f ◦ g ( x ). In fact, for certain choices of g ( x ), we prove that f ( x ) is monogenic if and only if any prime dividing disc( f ( x )) does not divide the index [