Any $n\times n$ matrix $A$ over a field $\mathbb{k}$ admits a rational canonical form (also called a Frobenius normal form).
The standard proof that this form exists appears to be to first prove a more general structural theorem for finitely generated modules over a principal ideal domain (which is proven, for example, in Section 12.1 of Dummit & Foote) and then view the action of $A$ as a $\mathbb{k}[x]$ module over a vector space.
My question: is there a simpler proof for the existence of the rational canonical form, which does not involve going through the structural theorem for modules over principal ideal domains? Also, are there other (conceptually different) proofs for the structural theorem besides the one presented in the Dummit & Foote?