Group action on Finite Field

Question

Suppose $F=\mathbb{F}_{p^n}$ is a degree-$n$ extension of $\mathbb{F}_p$. My questions concerns the action of the multiplicative group $(\mathbb{F}_p)^{\times}$ on $F$ by left multiplication. we can write $F=\mathbb{F}_p[x]/\langle \pi\rangle$ for some monic irreducible polynomial $\pi\in \mathbb{F}_p[x]$ of degree $n$. We may identify each element $\alpha\in F$ with a polynomial of degree strictly less than $n$. In this way, we can learn anything about the action of $(\mathbb{F}_p)^{\times}$ on $F$ by looking at the action of $(\mathbb{F}_p)^{\times}$ on $(\mathbb{F}_p)^n$. Is there a straightforward way to count the number of orbits? How about classifying each orbit?

Answer 1

$\newcommand{\FF}{\mathbb{F}}$ Yes the actions are isomorphic. An $\FF_p$-basis for $F$ is given by $1,\alpha,\alpha^2,\ldots,\alpha^{n-1}$, where $\alpha$ is a root of some degree $n$ irreducible polynomial $x^n + a_{n-1}x^{n-1} + a_1x + a_0$ over $\FF_p$. The action of $c\in\FF_p^\times$ on any basis element $\alpha^i$ is none other than $\alpha^i\mapsto c\alpha^i$. It's clear from this that $F$ and $(\FF_p)^n$ are isomorphic as vector spaces with $\FF_p^\times$-action.

Since $F$ is an integral domain, all orbits of $\FF_p^\times$ on $F$ either have size 1 (the orbit of $0\in F$), or have size $|\FF_p^\times| = p-1$ (for any other case). The number of orbits on $F^\times$ thus must be $|F^\times|/|\FF_p^\times| = (p^n-1)/(p-1) = p^n + p^{n-1} + \cdots + p + 1$.

The orbit containing $(c_1,\ldots,c_n)$ in $(\FF_p)^n$ just looks like $$\{(c_1,\ldots,c_n),(2c_1,\ldots,2c_n),\ldots,((p-1)c_1,\ldots,(p-1)c_n)\}$$