Suppose $A$ is a finite dimensional central simple algebra over a field $F$, $B$ is a simple subalgebra of $A$. $C$ is the centralizer of $B$ in $A$.
From Wedderburd-Artin theorem and the double centralizer theorem We know
- $A\cong M_n(\Delta_A)$，where $\Delta_A$ is a central division algebra over $F$
- $B\cong M_m(\Delta_B)$ ,where $\Delta_B$ is a division algebra over $F$
- $C\cong M_k(\Delta_C)$ ,where $\Delta_C$ is a division algebra over $F$
so what is the relationship between $\Delta_A$, $\Delta_B$ and $\Delta_C$, and $m,n,k$?
if we write their schur index, the dimension of a division algebra over its maximal subfield, as $Ind_A$, $Ind_B$, $Ind_C$, what is the relationship between these three numbers?