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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 $\mathbb F_q$?

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up vote 3 down vote accepted

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 $A$.

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The residue fields are necessarily finite extensions $\mathbb F_{q^t}$ of $\mathbb F_q$, and all of these occur : take $A=\mathbb F_{q^t}$ !

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As a pedagogical remark, I find it interesting to notice that it is easier to answer your question if you do not know that your algebra is a product of local algebras: ignorance is bliss! – Georges Elencwajg Feb 16 '13 at 16:04

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