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 D = D_0 tensor_{F_0} F, where D_0 is a division algebra, the center F_0 of D_0 contains k and trdeg(F_0/k) = r. We show that every division algebra of odd degree n >= 5 can be reduced to at most (n-1)(n-2)/2 parameters. Moreover, every crossed product division algebra of degree n >= 4 can be reduced to at most (log_2(n) - 1)n + 1 parameters. Our proofs of these results rely on lattice-theoretic techniques.