Editing Octonion (section)

==Integral octonions==
There are several natural ways to choose an integral form of the octonions. The simplest is just to take the octonions whose coordinates are [[integer]]s. This gives a nonassociative algebra over the integers called the Gravesian octonions. However it is not a [[Order (ring theory)|maximal order]] (in the sense of ring theory); there are exactly seven maximal orders containing it. These seven maximal orders are all equivalent under automorphisms. The phrase "integral octonions" usually refers to a fixed choice of one of these seven orders.

These maximal orders were constructed by {{harvtxt|Kirmse|1924}}, Dickson and Bruck as follows. Label the eight basis vectors by the points of the projective line over the field with seven elements. First form the "Kirmse integers" : these consist of octonions whose coordinates are integers or half integers, and that are half integers (that is, halves of odd integers) on one of the 16 sets
:{{math|∅ (∞124) (∞235) (∞346) (∞450) (∞561) (∞602) (∞013) (∞0123456) (0356) (1460) (2501) (3612) (4023) (5134) (6245)}}
of the extended [[quadratic residue code]] of length 8 over the field of two elements, given by {{math|∅}}, {{math|(∞124)}} and its images under adding a constant [[modular arithmetic|modulo]] 7, and the complements of these eight sets. Then switch infinity and any one other coordinate; this operation creates a bijection of the Kirmse integers onto a different set, which is a maximal order. There are seven ways to do this, giving seven maximal orders, which are all equivalent under cyclic permutations of the seven coordinates 0123456. (Kirmse incorrectly claimed that the Kirmse integers also form a maximal order, so he thought there were eight maximal orders rather than seven, but as {{harvtxt|Coxeter|1946}} pointed out they are not closed under multiplication; this mistake occurs in several published papers.)

The Kirmse integers and the seven maximal orders are all isometric to the [[E8 lattice|{{math|''E''<sub>8</sub>}} lattice]] rescaled by a factor of {{frac|1|{{radic|2}}}}. In particular there are 240 elements of minimum nonzero norm 1 in each of these orders, forming a Moufang loop of order 240.

The integral octonions have a "division with remainder" property: given integral octonions {{mvar|a}} and {{math|''b'' ≠ 0}}, we can find {{mvar|q}} and {{mvar|r}} with {{math|''a'' {{=}} ''qb'' + ''r''}}, where the remainder {{mvar|r}} has norm less than that of {{mvar|b}}.

In the integral octonions, all left [[ideal (ring theory)|ideals]] and right ideals are 2-sided ideals, and the only 2-sided ideals are the [[principal ideal]]s {{mvar|nO}} where {{mvar|n}} is a non-negative integer.

The integral octonions have a version of factorization into primes, though it is not straightforward to state because the octonions are not associative so the product of octonions depends on the order in which one does the products. The irreducible integral octonions are exactly those of prime norm, and every integral octonion can be written as a product of irreducible octonions. More precisely an integral octonion of norm {{mvar|mn}} can be written as a product of integral octonions of norms {{mvar|m}} and {{mvar|n}}.

The automorphism group of the integral octonions is the group {{math|''G''<sub>2</sub>('''F'''<sub>2</sub>)}} of [[order (group theory)|order]] 12,096, which has a [[simple group|simple]] subgroup of [[index of a subgroup|index]] 2 isomorphic to the unitary group {{math|<sup>2</sup>''A''<sub>2</sub>(3<sup>2</sup>)}}. The isotopy group of the integral octonions is the perfect double cover of the group of rotations of the {{math|''E''<sub>8</sub>}} lattice.