On the Level of a Quaternion Algebra

Sums of squares in octonion algebras

“…(ii) Let ϕ be a quadratic form over k of dimension greater than or equal to (ii) The proof is completely analogous to the one given in [LM,1.4] and will be omitted here.…”

“…This in turn implies that the quadratic form 2 m × 1, −x is isotropic over k 0 (x)(α m ), a contradiction to [LM,2.2].…”

Bounds on the Levels of Composition Algebras

“…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, It is possible to say more regarding the levels and sublevels of Q(n) and O(n) by employing the powerful machinery of Theorem 2.1. At a seminar in University College Dublin [4], Hoffmann kindly communicated his method of showing the existence of infinitely many quaternion algebras whose level is neither 2 k nor 2 k + 1 for some k. The family of quaternion algebras that he considers is {Q(n)}.…”

“…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 [12, theorem 2.5] and [15, theorem 3.11], with an ad-hoc argument proving the case where k = 1. Thus, the conjecture holds for l a power of two.…”

“…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.…”