I'm trying to solve a problem and I'm stuck.
Here is the original problem:
Let $A$ be a finite-dimensional algebra over a field $K$, such that for every $a\in A$, $a^7=a$. Show that $A$ is a direct product (sum?) of fields. What fields can arise?
We see that $A$ is Artinian and therefore its Jacobson radical is nilpotent. However from the fact that $a^7=a$ we see that there are no nilpotents, so Jacobson radical is zero. Therefore $A$ is semisimple and is a direct product of a matrix rings over division algebras. Since there are no nilpotents all matrix rings are 1-dimensional, so $A$ is a direct product of division rings.
Now we have to prove that all these division rings are fields. And that's where I am stuck. Can you give a hint what to do next? If I can prove that these division rings are finite I'm done, but I don't know how.