$f(x)$ is a primitive polynomial of degree $m$ over a field $\mathbf F_{p^m}$. As such, $\mathbf F_{p^m}[x]/f(x)$ forms a ring, with addition and multiplication ($\mkern-4mu\bmod f(x)$), and furthermore is a field since $f(x)$ is irreducible. Call it $F$
Now, replace $f(x)$ with $g(x)$, a non-primitive, non-irreducible polynomial of degree $m$ over $\mathbf F_{p^m}$ and where $g(0) = 1$. Call it $G$. I am having trouble finding information on line or referred to about the characteristics of such a $g$ or the algebraic structure of $G$ (a ring?). This is probably because I don't know the right terminology and keywords regarding these to search by. Fields and Rings are new to me.
Some questions I have are (assume $m > 2$):
- There are a finite number of elements in $G$ (because it is modulo $g$), but it doesn't form a field. What is it called? A "finite-ring"?
EDIT removed prior questions about cycle.
- For any non-zero $\alpha \in F$, $\exists n > 0$ such that $\alpha^n = 1$. This is not true for all $\alpha \in G$. Can we easily determine the ones that do have such an $n$?
- For any non-zero $\alpha \in F$, the powers of $\alpha$ form a closed group under multiplication. Again, not true $G$. In $G \exists \alpha$ whose powers never equal 1, so they cannot form a group (no multiplicative identity). And when $g(x)$ is allowed to have factors with multiplicity > 1, $\exists \alpha \in G$ such that $\alpha^k = 0, k \ge |\alpha|$. However, there $\exists \alpha \in G$ that do have powers that form cyclic multiplicative groups. Can we easily determine which ones they are?
Can you point me to a link or a paper (or even just the proper terms to use for searching) that discusses more about this non-irreducible case?