Monogenity of composition of quadrinomials

Abstract An irreducible monic polynomial f (x) ∈ Z x f (x) Zx is said to be monogenic if Z θ Z is the ring of integers of the number field Q (θ) Q () 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) Zx be such that f (g (x) ) is irreducible over Q Q. In this article, we establish a criterion that characterises the primes dividing the index, Q (α): Z [ α ] Q (): Z[ ] 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 [