# Orbits of Natural Galois Group Action

Let $m\mid n$ and let $G$ be the Galois group of $\mathbb{F}_{p^n}$ over $\mathbb{F}_{p^m}$. Write $G=\langle \pi^m\rangle$, where $\pi(t)=t^p$ is the Frobenius automorphism. I want to understand the orbits of the natural action of $G$ on $X=\mathbb{F}_{p^n}$ by automorphisms. First off, the total number of orbits should be given by applying Burnside;$$\frac{m}{n}\sum_{k=1}^{n/m} |X^k|,\quad (\ast)$$ where $X^k=\{\alpha \in \mathbb{F}_{p^n}: \pi^{mk}(\alpha)=\alpha\}$. I've managed to convince myself that the total number of orbits is given by the sum $$\frac{m}{n}\Big( p^{n}+ \varphi\big(n/m\big)p^{m} +\sum_{\substack{ m<d<n \\ d\mid n }} \mu_{n,m}(d) p^{n/d}\Big),$$ where $\mu_{n,m}(d)=\big|\{k: mk\leq n\mbox{ and } mk\equiv 0 \,(\mbox{mod }d)\}\big|$. I got this essentially by partitioning the right-hand side of the sum ( $\ast$) into three cases: $(mk, n)=m$, $(mk, n)=n$ and $(mk,n)\neq n,m$.

Question 1. The function $\mu_{n,m}$ I'm sure appears in other context. I've never really studied number theory much at all. But I was hoping somebody might point me to other contexts where this might appear.

Question 2. I'd like a suggestion about how to go about describing these orbits. I understand many of them will be singletons.But how can I think about the orbits that contain more than one element?

• So there is a one-to-one correspondance $$\{\mbox{orbits of Gal}(\mathbb{F}_{p^n}/\mathbb{F}_{p^m})\mbox{ on }\mathbb{F}_{p^n})\}\longleftrightarrow \{\pi(x)\in \mathbb{F}_{p^m}[x] \mbox{ monic and irreducible of degree }n\} ?$$ Can you elaborate, I'm a bit confused. – Andrew Nov 27 '15 at 6:13
• Let $\pi(x)\in \mathbb{F}_{p^m}[x]$ be monic and irreducible of degree $d\mid n$. If $\alpha$ is a root in some extension of $\mathbb{F}_{p^m}$, then $\pi(x)=(x-\alpha)(x-\alpha^{p^m})(x-\alpha^{p^{2m}})\cdots (x-\alpha^{p^{(d-1)m}})$ in its splitting field $\mathbb{F}_{p^m}[x]/\langle \pi(x)\rangle\cong \mathbb{F}_{p^{dm}}$. So I suppose in this way, we can obtain the corresponance between orbits of the action of $\mbox{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_{p^m})$ with the set of monic irreducible polynomials of degree $d$ in $\mathbb{F}_{p^m}[x]$ such that $d\mid n$ ? – Andrew Nov 27 '15 at 7:02