Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $\mathbb F_q$?

share|improve this question

2 Answers 2

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 $A$.

share|improve this answer

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

share|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.