Levels of division algebras

Abstract

Introduction. In [7] the level, sublevel, and product level of finite dimensional central division algebras D over a field F were calculated when F is a local or global field. In Theorem 1.4 of this paper we calculate the same quantities if all finite extensions K of F satisfy u(/C)<2, where u is the Hasse number of a field as defined in Section 1 deals with those properties of formally real fields that are useful in calculating levels of division algebras. In Sections 2 and 3 we restrict attention to the case of quaternion division algebras. Additional background to the problem of calculating levels of division algebras may be found in the introduction to [7]. The main references for Sections 2 and 3 are [9,10].We use standard terminology from the theory of quadratic forms and ordered fields as found in [6] and [11]. Let F x denote the nonzero elements of F. We shall assume throughout that char F =£ 2. We let D F (q) denote the nonzero elements of F represented by a quadratic form q over F. The topological space of orderings of a field F is denoted X F . Basic properties of X F and basic results on SAP fields can be found in [3] and [11].

Bounds on the Levels of Composition Algebras

“…For example, s (Q(15)) and s (Q(15)) ∈ [12,15] as a consequence of Theorem 3.9. However, 1 ⊥ 12 × T Q(15) is isotropic over F ( 1 ⊥ 15 × T P ), since it is a subform of 16 × T Q(15) of codimension 15, where i W 16 × T Q(15) ≥ 16 by [1, theorem 1.4].…”

Bounds on the Levels of Composition Algebras

“…For certain values of n, the lower bounds obtained for s (Q(n)) , s (Q(n)) , s (O(n)) and s (O(n)) in the above results are actually optimal with respect to the standard isotropy tests. For example, s (Q(15)) and s (Q( 15)) ∈ [12,15] as a consequence of Theorem 3.9. However, 1 ⊥ 12 × T Q( 15) is isotropic over F ( 1 ⊥ 15 × T P ), since it is a subform of 16 × T Q(15) of codimension 15, where i W 16 × T Q(15) ≥ 16 by [1, theorem 1.4].…”

Levels and sublevels of composition algebras

“…If (2 k + 1) × 1 ⊥ (2 k − 1) × a, b, −ab is isotropic over F , then s(D) ≤ 2 k by [L,Theorem 2.2], which is a contradiction. Clearly 1 ⊥ 2 k × a, b, −ab isotropic over F also implies that s(D) ≤ 2 k .…”