5 J an 2 00 0 LATTICES AND PARAMETER REDUCTION IN DIVISION ALGEBRAS

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 ≃ D0 ⊗F 0 F , where D0 is a division algebra, the center F0 of D0 contains k and trdeg k (F0) = r. We show that every division algebra of odd degree n ≥ 5 can be reduced to ≤ 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.