The third generator is a Hurwitz integer

Proved

HurwitzQ.basisK_mem_hurwitzIntegersQ

by jawneeboy · Sep 18, 2026 · Mathlib 0df444a (Lean v4.33.1)

algebrahurwitz-integersnumber-theoryquaternions

Let $\mathcal{H}$ be the Hurwitz subring of the rational quaternions: the four coordinates of an element are either all in $\mathbb{Z}$ or all in $\mathbb{Z}+\frac12$ . Put $\omega=(1+i+j+k)/2$ , where $i,j,k$ are the standard quaternion units.

k\in\mathcal{H}.

This records membership of a standard generator in the coordinate-defined subring.

Preamble

import Definitions.Def_HurwitzQ_hurwitzIntegersQ
import Definitions.Def_HurwitzQ_omega
import Mathlib.Algebra.Quaternion
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Tactic.Ring

open Quaternion QuaternionAlgebra HurwitzQ

Formal statement

theorem HurwitzQ.basisK_mem_hurwitzIntegersQ : ((Basis.self ℚ).k : ℍ[ℚ]) ∈ hurwitzIntegersQ := by sorry

Source

Standard definition: John H. Conway and Derek A. Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, A K Peters, 2003, §5.1, The Hurwitz Integral Quaternions. https://www.routledge.com/On-Quaternions-and-Octonions/Conway-Smith/p/book/9781568811345 The displayed assertion is an elementary consequence of this definition; no numbered theorem attribution is claimed.

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