Chapter 38: The Elliptic Quaternions

the relevant chapter previewed the quaternions ℍ_τ = ℝ_τ[i,j,k] as a non-commutative extension beyond ℂ_τ (Remark [rem:ch39-quaternions]). This chapter carries out the full construction: ℍ_τ is the first non-commutative algebraic structure earned from the τ axioms. Extending ℂ_τ to four real dimensions while preserving the division algebra property forces the sacrifice of commutativity. We prove that every nonzero quaternion has a multiplicative inverse (the relevant theorem, I.T44), establish the embedding tower ℝ_τ ⊂ ℂ_τ ⊂ ℍ_τ, and preview the Hurwitz classification (I.R22).