Galois Groups of Powered Quadratic Unit Polynomials: Real Kummer Structure and Imaginary Cyclotomic Classification

Abstract

This preprint develops a general Galois-theoretic and arithmetic theory for power-compositional polynomials attached to quadratic units. Let F/ℚ be a quadratic field, let ε ∈ O_F× be a nonrational quadratic unit with conjugate ε̄, and let ℓ ≥ 2. The central family is Qε,ℓ(x) = (x^ℓ − ε)(x^ℓ − ε̄) = x^(2ℓ) − Tr_F/ℚ(ε)x^ℓ + N_F/ℚ(ε). The paper determines the splitting-field structure of this family for arbitrary exponents and separates the fundamentally different arithmetic behavior of real and imaginary quadratic units. For real quadratic units, the natural base field is the compositum of the quadratic field F with the cyclotomic field required by the norm and parity of ℓ. The radical extension is governed by the order rε,ℓ of the Kummer class of ε. This gives an exact splitting-field degree formula, explicit factorization when rε,ℓ < ℓ, and a compatible-pair description of the resulting Galois group. At full Kummer order, the norm-positive and odd-exponent branches split, while the norm-negative even branch exhibits a parity obstruction. Fundamental units are shown to have full Kummer order at every prime exponent. The paper also determines the polynomial discriminant, distinguishes the quadratic-order index [O_F : ℤ[ε]] from the root-order indices occurring in monogenicity problems, and gives the complete finite ramification support of the splitting field, including the exceptional dyadic behavior. The golden-ratio family ε = φ = (1 + √5)/2 is developed as a self-contained model for the general real-quadratic theory. Its explicit generators, affine Galois action, stabilizers, complement structure, discriminant, ramification, and splitting behavior provide a concrete prototype for the subsequent Kummer formulation. For imaginary quadratic fields, Dirichlet's unit theorem forces every unit to be torsion. The radical problem therefore collapses to cyclotomic arithmetic. The paper gives a complete classification of the resulting splitting fields and Galois groups and develops explicit Gaussian and Eisenstein cases, including factorizations, root-orbit decompositions, field-generation criteria, and the distinction between generation by an individual root and generation by the full root set. The resulting framework gives a unified description of how quadratic-unit arithmetic controls power-compositional splitting fields: infinite-order real units lead to genuine Kummer extensions and splitting obstructions, whereas torsion imaginary units lead to purely cyclotomic extensions. (ARCHIVED PRIOR WORKING DRAFT DESCRIPTION) We provide a complete analysis of Galois groups arising from polynomials Q(x) = x²ˡ − Tr(ε)xˡ + N(ε) where ε is a unit in a quadratic field. For general real quadratic units with N(ε) = ±1, we prove the splitting field has degree 2ℓ·φ(ℓ) with the Galois group structure: (Z/ℓZ ⋊ (Z/ℓZ)×) ⋊ Z/2Z This is a semidirect product where the complex conjugation acts by inversion. We further derive the exact discriminant formula via the resultant method: disc(Q) = ℓ²ˡ · N(ε)ˡ⁻¹ · Δˡ For imaginary quadratic units, the theory collapses: all such units are roots of unity, reducing the problem to standard cyclotomic fields with abelian Galois groups and degrees φ(n). This dichotomy is explained by Dirichlet's Unit Theorem: Real quadratic fields have unit rank 1 (infinite units), allowing for the geometric growth of the field extensions. Imaginary quadratic fields have unit rank 0 (finite torsion only), resulting in the collapse to cyclotomic theory.