Let $K$ be a number field, $Z\_K$ its ring of integers, and $p$ a rational prime number. Then $A\_p = Z\_K/(p)$ is a finite ${\mathbb F}\_p$-algebra. Using ideal arithmetic in $Z\_K$ and the Chinese remainder theorem it is easily checked that $A\_p$ is the direct sum of finite fields if $p$ is unramified, and has additonal nil-rings as components if $p$ is ramified. 


This decomposition result looks simple enough to have a direct and not too complicated proof. Basing it on the close relation between $A\_p$ and ${\mathbb F}\_p[X]/(f)$, where $f$ is the minimal polynomial of a generator of $K$, brings in problems with primes dividing the discriminant of $f$. Thus let me state my main question explicitly:


**Is there a simple proof that $A\_p$ is a direct sum of finite fields and some easily described nil rings?**


In addition, I'd be grateful for pointers to the relevant literature, in particular to classification theorems of which the result above is a special case.



Decomposition of finite algebras over finite fields
The general form of the answer is easy to anticipate: a finite commutative ring is artinian. It will be a product of finite local rings, of characteristic that is a prime number. So your question is really which local rings you get in this case. I believe a certain amount can be done by linear algebra.


