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.