M. Lorenz, Z. Reichstein, Lattices and parameter reduction in division algebras , January 2000.

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 = D0 tensor F0 F, where D0 is a division algebra, the center F0 of D0 contains k and trdegk(F0) = r.

We show that every division algebra of odd degree n 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 <= ([log2(n)] -1) n + 1 parameters.

Our proofs of these results rely on lattice-theoretic techniques.

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