Lattices and parameter reduction in division algebras
Martin Lorenz and Zinovy Reichstein
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 $D \simeq D_0 \otimes_{F_0} F$, where $D_0$ is a central simple algebra of degree $n$, the center $F_0$ of $D_0$ a division algebra, the center $F_0$ of $D_0$ contains $k$ and ${\rm trdeg}_k(F_0) = r$.
We show that every division algebra of odd degree $n \geq 5$ can be reduced to $\leq {\frac{1}{2}}(n-1)(n-2)$ parameters. Moreover, every crossed product division algebra of degree $n \geq 4$ can be reduced to $ \leq (\lfloor \log_2(n) \rfloor - 1)n + 1$ parameters. Our proofs of these results rely on lattice-theoretic techniques.