LATTICES AND PARAMETER REDUCTION IN DIVISION ALGEBRAS

MARTIN LORENZ ZINOVY REICHSTEIN

Abstract. Let k be an algebraically closed field of characteristic 0 and let D be a division algebra whose center F contains k. We shall say that D can be reduced to r parameters if we can write D ≃ D_{0} ⊗_{F_{0}} F, where D_{0} is a division algebra, the center F_{0} of D_{0} contains k and trdeg_{k}(F_{0}) = r.

We show that every division algebra of odd degree n ≥ 5 can be reduced to ≤ \frac{1}{2}(n − 1)(n − 2) parameters. Moreover, every crossed product division algebra of degree n ≥ 4 can be reduced to ≤ (⌊log_{2}(n)⌋ − 1)n + 1 parameters. Our proofs of these results rely on lattice-theoretic techniques.