Iterates of Quadratics and Monogenicity

We investigate monogenicity and prime splitting in extensions generated by roots of iterated quadratic polynomials. Let f (x) be an irreducible, monic, quadratic polynomial, and write fⁿ (x) for the n^th 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.