Sums of Squares in Certain Quaternion and Octonion Algebras

Formulae for the levels and sublevels of certain quaternion and octonion algebras are established. Corollaries concerning the equality of levels and sublevels of quaternion algebras with those of associated octonion algebras are presented. Let R be a not necessarily associative ring with unity. The level and sublevel of R, respectively denoted by s(R) and s(R), are defined as follows: s(R) = inf{n ∈ N | there exist x1, . . . , xn ∈ R such that ∑n t=1 xt 2 = −1}, s(R) = inf{n ∈ N | there exist x1, . . . , xn+1 ∈ R \ {0} such that ∑n+1 t=1 xt 2 = 0}. Let F be a field of characteristic different from 2. For a, b ∈ F×, the quaternion algebra (a, b)F is a 4-dimensional F -vector space with basis {1, i, j, k} satisfying i = a, j = b and ij = −ji = k. For a, b, c ∈ F×, the octonion algebra (a, b, c)F is isomorphic to (a, b)F ⊕ (a, b)F e, where e = c, with its multiplication being determined by (u1, v1)(u2, v2) = (u1u2 + cv2v1, v2u1 + v1u2), where u1, u2, v1, v2 ∈ (a, b)F (here, denotes conjugation). The related problems of determining the numbers attainable as the levels and sublevels of quaternion and octonion algebras remain open, and motivate our investigations. Given a, b ∈ F×, we study whether the level (respectively, the sublevel) of (a, b)F equals that of (a, b, x)F ((x)). We will provide a partial answer to these questions, by showing that the respective equalities hold whenever the level or sublevel of (a, b)F belongs to an associated family of intervals. Moreover, we will show that these equalities always hold for a particular class of quaternion algebras, conjectured to contain members of level and sublevel n for all n ∈ N (see [4]). Throughout, we will employ standard concepts and notation regarding quadratic forms. Our notation coincides with that employed in [3], aside from our usage of n×φ to denote the orthogonal sum of n ∈ N copies of a quadratic form φ. Moreover, for a, b ∈ F×, we will let k(a) (respectively k(a, b)) denote the least n ∈ N such that n× 〈1,−a〉 (respectively n× 〈1,−a,−b, ab〉) is isotropic (over F , unless stated otherwise). If such an n exists, then n = 2 + 1 for some k ∈ Z. Otherwise, the quantity is said to be infinite. In order to obtain the aforementioned results, we will establish characterisations of the levels and sublevels of certain quaternion and octonion algebras, namely those with “transcendental parameters”. These characterisations provide analogues of a theorem of Tignol and Vast (see [5]), the statement of which is included in the following result. Theorem 1. Let a, b ∈ F×, Q = (a, x)F ((x)) and O = (a, b, x)F ((x)). Then (a) s(Q) = min {s(F ( √ a)), k(a)} and s(Q) = min {s(F ( √ a)), k(a)− 1}, (b) s(O) = min {s ((a, b)F ) , k(a, b)} and s(O) = min {s ((a, b)F ) , k(a, b)− 1}. 1