Hello everybody


There is a nice classical result in linear algebra: if $A, B$ are two matrices in $M\_n(k),$ where $k$ is a field, and $B$ commutes with every element of $M\_n(k)$ which commutes with $A$, then $B = f(A)$ for some polynomial $f(x)$ in $k[x].$


I was wondering if anybody knows any (important) theorem which is proved using this result. Thank you.



Looking for applications of a nice result in linear algebra
Tate's famous "Endomorphisms of Abelian Varieties over Finite Fields," which proves the Tate conjecture in the finite field case, uses the full force of the theorem of bicommutation in a reduction lemma. As KConrad mentions in the comments, the result you've cited is the special case of this theorem where one works with the subalgebra generated by one element.


