Math 2.0

Number theory Object

Hurwitz quaternions

Builds on The quaternionsQuaternion algebraOrder (ring theory)Lagrange's four-square theorem

Description

Let ℍ\mathbb{H} be the quaternions, with basis 1,i,j,k1, i, j, k over ℝ\R and i2=j2=k2=ijk=−1i^2 = j^2 = k^2 = ijk = -1. The Hurwitz quaternions are the elements a+bi+cj+dk∈ℍa + bi + cj + dk \in \mathbb{H} such that either a,b,c,da, b, c, d are all in ℤ\Z or all in ℤ+12\Z + \tfrac{1}{2}. They form a ring, written ℋ\mathcal{H} here. Equivalently, ℋ\mathcal{H} is the ℤ\Z-span of ii, jj, kk and ω=12(1+i+j+k)\omega = \tfrac{1}{2}(1 + i + j + k).

Remarks

The ring ℋ\mathcal{H} is a maximal order in the rational quaternion algebra (−1,−1)ℚ(-1, -1)_{\Q}. It contains the Lipschitz quaternions, those with a,b,c,d∈ℤa, b, c, d \in \Z, as a subring of index 22; the Lipschitz quaternions are an order but not a maximal one. The reduced norm a2+b2+c2+d2a^2 + b^2 + c^2 + d^2 takes integer values on ℋ\mathcal{H}; for example ω\omega has norm 4⋅14=14 \cdot \tfrac14 = 1. The units of ℋ\mathcal{H} are the 2424 elements ±1,±i,±j,±k\pm 1, \pm i, \pm j, \pm k and 12(±1±i±j±k)\tfrac12(\pm 1 \pm i \pm j \pm k); they form the binary tetrahedral group. Hurwitz quaternions admit division with remainder on each side, so every one-sided ideal of ℋ\mathcal{H} is principal. This is a standard route to Lagrange's four-square theorem.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

Show the path as a graph
Euler's four-square identity, statement, no evidence levelEuler's four-square i…statementGroup, concept, no evidence levelGroupconceptOrder (ring theory), concept, no evidence levelOrder (ring theory)conceptQuaternion algebra, concept, no evidence levelQuaternion algebraconceptLagrange's four-square theorem, statement, no evidence levelLagrange's four-squar…statementRing, definition, no evidence levelRingdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionDivision ring, definition, no evidence levelDivision ringdefinitionThe quaternions, example, no evidence levelThe quaternionsexampleHurwitz quaternions, object, no evidence levelHurwitz quaternionsobject
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euler's four-square identity statement
  2. Group concept
  3. Order (ring theory) concept
  4. Quaternion algebra concept
  5. Lagrange's four-square theorem statement
  6. Ring definition
  7. Unit of a ring definition
  8. Division ring definition
  9. The quaternions example
  10. Hurwitz quaternions object

Further reading

  1. Web https://en.wikipedia.org/wiki/Hurwitz_quaternion
About this page
TagQHTAEPTKEB
URIm2://QHTAEPTKEB
Kindobject
WritingW0 stub
MSC11R52, 16H10
Instance ofMaximal order
UsesThe quaternions, Quaternion algebra, Order (ring theory), Lagrange's four-square theorem
Same as
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.