$\forall A,B \in M_n(\mathbb F),AB=BA,\exists C \in M_n(\mathbb F),f(x),g(x)\in \mathbb F[x],A=f(C),B=g(C)?$

Question

Question:$\forall A,B \in M_n(\mathbb F),AB=BA,\exists C \in M_n(\mathbb F),f(x),g(x)\in \mathbb F[x],A=f(C),B=g(C)?$ Or:$\forall \sigma,\tau \in L(\mathbb F),\sigma \tau=\tau \sigma,\exists \upsilon \in L(\mathbb F),f(x),g(x) \in \mathbb F[x],\sigma = f(\upsilon),\tau=g(\upsilon)?$

I find if $A=f(C),B=g(C)$,then $AB=BA$.
I wonder if the converse is true,but I don't know how to proove it.
Help please~