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

This paper is the first autonomous article extracted from a broader "saga" , still under investigation called "remanent tomography of multiplicative structures". The original programme studies how arithmetic information is retained by finite critical orbits when modular levels are compared. In the quadratic family fc(x)=x²+cf_c(x)=x²+c, the passage from a prime pp to p2p^2 naturally isolates the multiplier of the terminal cycle as the key multiplicative datum governing the lift. This led to the algebraic branch developed here. The paper constructs explicit multiplier towers attached to critical orbits accessible from the marked point 00. For the fixed-point branch, it introduces polynomial families QnQ_n and ΦnΦ_n, proves a quadratic recursion, a norm identity, irreducibility over QQ, and an exact resultant relation. The study is then extended to two accessible period-33 packets, with preperiods 22 and 33. Their multiplier polynomials have degrees 66 and 1212, and their Galois groups are shown to be S6S_6 and S12S₁₂, respectively. The main result concerns the associated 22-primary radical extensions. Under a general symmetric-group criterion, the splitting fields of Ψ(X2s)Ψ(X2^s) are shown to attain the maximal possible degree at every depth. Both explicit packets satisfy this criterion, yielding complete maximal Kummer towers. These results isolate and complete one algebraic component of the wider programme. The finite-field existence, localisation, lifting and reconstruction questions that motivated the construction remain separate problems and are intended to be treated in companion works.