# Commutative Algebra over a Finite Field

Let $A$ be a commutative algebra over the finite field $\mathbb F_q$, of order $q$, where $q=p^l$, for a prime $p$ and a positive inteher $l$. Assume that $A$ is finite-dimensional. So, $A$ is a finite ring. Hence, we have the decomposition

$$A=R_1\oplus\ldots\oplus R_k,$$

where each $R_i$ is alocal ring. What can be said about the residue fields of ecah local ring $R_i$ ? Are all equal to $\mathbb F_q$?

What about the "trivial" case where $A$ is a finite field extension of $\mathbb F_q$? Then $A = R_1$ is local with residue field $A$.
The residue fields are necessarily finite extensions $\mathbb F_{q^t}$ of $\mathbb F_q$, and all of these occur : take $A=\mathbb F_{q^t}$ !