Multiplier towers of the quadratic family x² + c: irreducibility, Galois groups, and maximal Kummer towers

Abstract

This paper isolates and settles one algebraic component of a broader programme on how arithmetic information is retained by finite critical orbits across modular levels. This program called "Remanent tomography of multiplicatives structures" is currently in progress. In the quadratic family f(x) = x^2 + c, the passage from a prime p to p^2 is governed by an affine holonomy along the cycle, whose additive part is a coboundary; any obstruction is therefore carried by the multiplier of the terminal cycle. This motivates the study of multipliers as algebraic objects over the rationals, which is the subject of the paper. We first recall and set up the tower attached to critical orbits landing on a fixed point: parameter polynomials Q_n and multiplier polynomials Phi_n, both of degree 2^(n-1) - 1. We establish a quadratic recursion for Phi_n, a norm identity relating the two towers, and an exact resultant identity showing that the product of the multipliers along any level is, up to sign, a power of 2. Irreducibility of these towers is due to Buff, Epstein and Koch, going back to Gleason; we recall it and note the form the Eisenstein condition takes in the multiplier coordinate. We then treat two accessible packets of terminal period 3, with preperiods 2 and 3. Their multiplier polynomials, of degrees 6 and 12, are computed explicitly, and their Galois groups over the rationals are shown to be the full symmetric groups S_6 and S_12. The main result concerns the associated 2-primary radical extensions. We give a general criterion, for a monic irreducible polynomial with symmetric Galois group whose constant term is plus or minus an odd power of 2 and whose discriminant field is ramified at an odd prime, under which the splitting field of Psi(X^(2^s)) attains the largest degree compatible with that constant term, at every depth s. Both packets satisfy the criterion, so their entire Kummer towers are maximal: no hidden multiplicative relation among the multipliers ever appears. A corollary records the converse constraint: the resultant identity forces every such family to contain radicals of 2, so two families of this type are never linearly disjoint. Existence, localisation, lifting and reconstruction questions over finite fields, which motivated the construction, are separate problems treated elsewhere. All polynomial identities were verified by symbolic computation. The author would welcome pointers to overlapping literature.