Affine Galois Realizations from Real Quadratic Units: Kummer Defects, Composita, and Radical Lattices

Abstract

This preprint studies explicit Galois extensions arising from radicals of units in real quadratic fields. Let F/ℚ be a real quadratic field, let ε ∈ O_F× be a nonrational unit, and let ℓ ≥ 2. The associated trace–norm polynomial is Qε,ℓ(x) = (x^ℓ − ε)(x^ℓ − ε̄), where ε̄ denotes the nontrivial quadratic conjugate of ε. The paper develops the splitting fields, Galois groups, cyclotomic intersections, radical subfields, composita, ramification, and discriminant structure of this family. The general splitting field is controlled by two independent arithmetic phenomena: the order of the Kummer class of ε after adjoining the required roots of unity, and the intersection of the quadratic field F with the relevant cyclotomic field. This yields exact degree formulas and cyclic-kernel descriptions of the corresponding Galois groups. At full Kummer order, the splitting behavior is determined by the norm of ε and the parity of ℓ; in particular, the norm-negative even-exponent case exhibits a genuine nonsplitting obstruction that persists across all possible quadratic–cyclotomic overlap configurations. For fundamental units and prime exponents, the Kummer class has full order. This produces explicit affine Galois realizations and an exact dichotomy governed by the cyclotomic intersection. In the generic prime case the Galois group is AGL(1,ℓ) × C₂, while the unique quadratic–cyclotomic overlap removes the additional C₂ factor and leaves AGL(1,ℓ). The construction gives infinitely many pairwise-distinct explicit Galois extensions over ℚ with controlled solvable group structure and ramification. The paper also studies finite composita of the splitting fields. For a fixed quadratic field and unit, composita indexed by several exponents collapse to the field associated with their least common multiple, and the failure of linear disjointness is measured by an explicit fiber-product index. This gives a structured hierarchy of metabelian Galois extensions rather than independent radical layers. A parallel analysis is carried out for the non-Galois radical fields Eₙ = F(ε^(1/n)). For a fundamental unit ε, these fields have exact degree n over F, admit explicit normal closures, and satisfy the divisor-lattice identities Eₐ ∩ Eᵦ = E_gcd(a,b) and EₐEᵦ = E_lcm(a,b). Their automorphism groups, signatures, discriminant bounds, ramification support, normality, and abelian-containment properties are determined explicitly. In particular, Eₙ/F is Galois and abelian exactly when n = 2; for n > 2 the fields form an explicit family of nonnormal radical extensions generated by algebraic units. The resulting framework connects Kummer theory, cyclotomic arithmetic, quadratic units, affine Galois groups, radical-field lattices, ramification, and class-field-theoretic boundaries within a single explicit family. (Archived Description of Previous Working Draft) We prove that for every integer ℓ ≥ 2 and every real quadratic unit ε with gcd(ℓ, Δ) = 1, the group (Z/ℓZ ⋊ (Z/ℓZ)×) ⋊ Z/2Z of order 2ℓ·φ(ℓ) occurs as a Galois group over the rationals Q. The realization is given explicitly by the splitting field of the monic polynomial: Q(x) = x²ˡ − Tr(ε)xˡ + N(ε) This provides a complete solution to the Inverse Galois Problem for this parametric family of affine semidirect products with a Z/2Z top group. Via Hilbert's Irreducibility Theorem, we show that each such group is realized infinitely often. These extensions provide explicit constructions of: Solvable extensions of arbitrarily large degree over Q with completely known Galois structure. Explicit cyclic abelian extensions of real quadratic fields generated by radicals of units, providing concrete examples of Class Field Theory. Note: This paper establishes the explicit construction of these groups using elementary methods involving Kummer theory and Dirichlet's unit theorem.