Let $A$ be a central simple algebra with a finite dimension over the field $F$.
Let $A \supset K \supset F$ be a subfield.
Show that $C_A(K)$ and $A \otimes_F K$ are both central simple algebras over the center K. and also $[C_A(K)] = [A \otimes_F K]$ in Brauer group $B(K)$
Well , I have a proof for the first part but for the last part, I have completely no idea..
I think it might be connected to the fact that $A \cong K \otimes_F C_A(K)$
Thanks !