TTwo Observations on the Cubic Field Q(α), α³-α-1=0: A New Analytic Invariant and a Geometric Classification of Brocard Solutions

We study the cubic field K = Q(α) defined by α³-α-1=0. Two independent observations are presented. First, we define a new analytic invariant of K, τ∞ = 1/|Γ(α₁)Γ(α₂)Γ(α₃)| ≈ 0.6334, where αᵢ are the conjugates of α. This constant does not appear in the literature and may be related to Mahler measures or regulators of K. Second, we give a geometric classification of the known solutions to Brocard's problem n!+1=m²: the solutions (4,5) and (5,11) arise from the elliptic curve y²=x³-x+1 (the norm curve of K), while (7,71) arises from the genus 3 curve ∏(x+k)+1 = y². The numbers 3 and 5 are the only prime orders of the icosahedral group A₅, suggesting a hidden symmetry. Both observations are elementary but appear to be new.