Sums of squares in octonion algebras

Abstract

Abstract. Sums of squares in composition algebras are investigated using methods from the theory of quadratic forms. For any integer m ≥ 1 octonion algebras of level 2 m and of level 2 m + 1 are constructed.

Bounds on the Levels of Composition Algebras

“…Indeed, for n = 2 k −1, we have that s(C) ≤ n if and only if 1 ⊥ n × T P is isotropic. This does not hold in general however, as is evidenced by Koprowski's [8] construction of a quaternion algebra of level 2 k such that 1 ⊥ 2 k × T P is anisotropic for all k, and Pumplün's analogous octonion algebra (see [18]). For n = 2 k , we do have the following equivalence however:…”

“…Since Q(n) and O(n) coincide with constructions of Laghribi and Mammone [10] and Pumplün [18] when n = 2 k + 1 for k ≥ 1, the level component of Conjecture 3.1 has been established for such values of n (see [10, theorem 2.1] and [18, theorem 3.1]). Indeed, both results extend to the case where k = 0 by Theorem 3.2.…”

“…In the case where l is a 2-power, Q represents a construction of Laghribi and Mammone (see [10]), with O coinciding with one of Pumplün (see [18]), allowing us to conclude that s(Q ) = s(O ) = 2 k . Moreover, s(Q ) = s(O ) = 2 k for k ≥ 2, by [12, theorem 2.5] and [15, theorem 3.11], with an ad-hoc argument proving the case where k = 1.…”

“…The composition algebras of dimension 2 are the quadratic étale F -algebras, while those of dimension 4 are the (non-commutative) quaternion algebras, and those of dimension 8 are the (non-commutative and non-associative) octonion algebras. For a, b ∈ F × , the quaternion algebra Q = a,b F over F is a 4-dimensional F -vector space with basis {1, i, j, k}, where i 2 = a, j 2 = b and ij = −ji = k. Analogously, for a, b, c ∈ F × , the octonion algebra a,b,c F over F is an 8-dimensional F -vector space with basis {1, i, j, k, e, ie, je, ke}, with multiplication, as described by the Cayley-Dickson doubling process (see [18]), satisfying i 2 = a, j 2 = b and e 2 = c.…”

“…Laghribi and Mammone recovered these values as the level of a quaternion algebra using function field techniques (see [10]). Pumplün employed this methodology in [18] to construct octonion algebras of level 2 k and 2 k + 1. By constructing a quaternion algebra of sublevel 3, Krüskemper and Wadsworth produced the first example of a quaternion algebra whose sublevel was not a power of two (see [9]).…”