Iterates of Quadratics and Monogenicity

We investigate monogenicity and prime splitting in extensions generated by roots of iterated quadratic polynomials. Let f(x)[x] be an irreducible, monic, quadratic polynomial, and write fⁿ(x) for the nᵗʰ iterate. We obtain necessary and sufficient conditions for fⁿ(x) to be monogenic for each n. We use this to construct multiple families where fⁿ(x) is monogenic for every $n>0$.