Affine Galois Closures, Ray-Class Quotients, and Radical-Field Ramification over Real Quadratic Fields

Abstract

This preprint develops the local and global arithmetic of affine Galois closures arising from radicals of units in real quadratic fields. Let F/ℚ be a real quadratic field, let ε ∈ O_F× be a fundamental unit, and let ℓ be an odd prime. The principal objects are the radical field E = F(ε^(1/ℓ)), its normal closure F(ε^(1/ℓ), ζℓ), and the corresponding trace–norm polynomial over ℚ. The paper determines the Galois closure and its affine group structure, including the unique quadratic–cyclotomic overlap in which the multiplier subgroup drops to the subgroup of quadratic residues. Intermediate fields and subgroup structures are described explicitly, together with the reciprocal pairing of roots and the associated block systems. A norm-one layer yields an additional dihedral structure linking the affine and quadratic components. The main emphasis is the local arithmetic of the radical extension. For primes above ℓ, the decomposition and inertia groups are determined in each possible local configuration. The analysis separates unramified, partially ramified, and totally wild behavior and computes the corresponding lower ramification filtration, local different exponents, and conductor data. These local invariants are then assembled into global discriminant formulas and ramification statements. For the degree-ℓ radical field E/F, the power basis generated by ε^(1/ℓ) is compared with the maximal order. The local power-basis index is related explicitly to the wild ramification break, producing the exact identity 2j_L = ℓ(eℓ − 1) + 1 − cℓtℓ, where j_L is the local index exponent and eℓ, cℓ, and tℓ encode the local ramification and different data. This gives a direct arithmetic relation between integral-basis failure and wild ramification. The paper also determines relative and absolute different and discriminant formulas, root discriminants, finite ramification support, local splitting behavior, and the arithmetic of conjugate radical fields. Ray-class quotients and Artin characters are used to locate the abelian portions of the construction and to distinguish them from the genuinely nonabelian affine closure. The resulting theory provides an explicit bridge between Kummer extensions, affine Galois groups, local ramification theory, discriminants, integral bases, and class field theory for a concrete family of radical extensions over quadratic fields. (Archived description for prior working draft) For every real quadratic field $K$ and every odd prime $\ell$ coprime to the discriminant, we construct an explicit polynomial Q(x) = x²ˡ − Tr(ε)xˡ + N(ε) whose splitting field realizes the affine semidirect product: G = (Z/ℓZ ⋊ (Z/ℓZ)×) ⋊ Z/2Z This paper provides a complete solution for this parametric family in the Inverse Galois Problem. We determine the precise arithmetic of the extension, proving that the cyclic subfield L is contained in the Narrow Ray Class Field Kᶠ⁺ modulo ℓ⋅∞₁∞₂. We further derive the exact discriminant formula for this family: disc(Q) = ℓ²ˡ ⋅ N(ε)ˡ⁻¹ ⋅ Δˡ The construction offers the first infinite, explicit family of such abelian extensions of real quadratic fields using only the fundamental unit.