S₃-Extensions of ℚ(i) from the Pythagorean Angle Lattice

An explicit infinite family of S₃-extensions of the Gaussian rationals, constructed geometrically from the cubic projection of the Pythagorean Angle Lattice of integer right triangles : We study the cubic projection branch of the Pythagorean Angle Lattice. For each primitive Gaussian integer W = u + vi ∈ ℤ[i]prim, the lattice gives a cubic polynomial: f_W(y) = y³ − 3γy² − 3y + γ = 0, γ = (u² − v²)/(2uv). We prove that f_W is irreducible over ℚ(i) if and only if W is not a cube in ℤ[i]. Consequently, the splitting field K_W of f_W over ℚ(i) has Galois group: Gal(K_W/ℚ(i)) ≅ { S₃ if W is not a cube in ℤ[i], ℤ/2 if W is a cube in ℤ[i]. } The exceptional set of cube values has natural density zero. We identify the dihedral Galois representation attached to the S₃ branch and compute its fixed field. This provides an explicit family of S₃-extensions of ℚ(i) arising from the geometry of integer right triangles.