# Finite local rings

There is some classification of finite commutative local rings. For example how many not isomorphic finite local rings with the same order $p^k$ and the same residue field $\mathbb F_p$ exist?

• You surely do not mean finite field there. – Mariano Suárez-Álvarez Jun 5 '12 at 15:57
• – Mariano Suárez-Álvarez Jun 5 '12 at 16:00
• Miguel, you can edit the text of the question (by clicking on the edit link that appears right below it) It i better if you fix it there, and then we can remove these comments. – Mariano Suárez-Álvarez Jun 5 '12 at 16:16
• Gracias Mariano – Miguel Jun 5 '12 at 17:10
• Martin, I do not understand your comment, you want to explain? – Miguel Jun 6 '12 at 9:41

Miguel, I S Cohen's structure theorems on complete local rings answers (more or less) your question, since a finite local ring is complete. Your ring $R$ must be a quotient of a ring of power series $k[[x_1,\dots, x_n]]$ where $k$ is either a finite field or the ring of Witt vectors over a finite field. You can take $n=m$ to be the minimal number of generators of the maximal ideal $\mathfrak{m}=(y_1,\dots,y_m)\subset R$, or $n=m-1$ if $R$ does not contain the residue field.
The answer to your specific question looks like more complicated in general, but you can try to play with the theorem to get an answer. For instance, the only examples with $p^2$ elements are simply $\mathbb{F}_p[t]/(t^2)$, $\mathbb{Z}/(p^2)$, and $\mathbb{F}_{p^2}$.