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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?

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An orbit of Frobenius is the same thing as a monic irreducible polynomial (namely the monic irreducible polynomial whose roots are the elements of the orbit). So the correct count of the number of orbits is given by the necklace polynomials, summed over all the possible degrees of irreducible polynomials.

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  • $\begingroup$ 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. $\endgroup$ – Andrew Nov 27 '15 at 6:13
  • $\begingroup$ 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 $d\mid n$ ? $\endgroup$ – Andrew Nov 27 '15 at 7:02
  • $\begingroup$ @Andres: yes, that's right. $\endgroup$ – Qiaochu Yuan Nov 27 '15 at 7:31

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