We define $\hat{\mathbb{Z}}$ as the inverse limit $\varprojlim \mathbb{Z}/n \mathbb{Z}$. One can also show that $\hat{\mathbb{Z}}$ is isomorphic to the absolute Galois group $G_k = \operatorname{Gal}(\bar{k}/k)$ of a finite field $k \simeq \mathbb{F}_q$. Literature mentions that there two generators of the absolute Galois group of any finite extension of $k$, namely:
- the arithmetic Frobenius automorphism $x \mapsto x^q$,
- the inverse of the arithmetic Frobenius automorphism, the so-called geometric Frobenius automorphism.
These two elements have infinite order, so the subgroup generated by them are isomorphic to $\mathbb{Z}$ which can be considered to be a subgroup of $\hat{\mathbb{Z}}$. Moreover, these two elements are so-called topological generators of $G_k$, but I do not know yet what this is supposed to mean. Anyway, I guess one of these mentioned facts is the reason we call the Frobenius elements canonical generators.
Questions: What exactly are topological/canonical generators? Are there non-canonical generators of $G_k$ resp. $\hat{\mathbb{Z}}$? If yes, how would they look like?
It would be also great if you could give me an example of such a generator. Thanks for helping!
You asked what a non-canonical generator might be. Here’s one answer. Remember that a topological generator $\lambda$ of a group $G$ is just an element that generates a dense subgroup of $G$. Thus, if $G$ is commutative, written additively, you’re asking for the closure of $\Bbb Z\lambda$ to be $G$.
If you think of $\hat{\Bbb Z}$ as the direct product $\prod_p\Bbb Z_p$, then the Frobenius can be thought of as the $\infty$-tuple $(1,1,1,\cdots)$, all componentes being $1$. Its inverse has all components equal to $-1$. But each factor $\Bbb Z_p$ has a veritable profusion of (topological) generators, in fact any (multiplicative) unit of $\Bbb Z_p$ is a topological generator of the additive group $\Bbb Z_p$.
So, I guess that the topological generators of $\hat{\Bbb Z}$ are the $\infty$-tuples where the $p$-th component is a unit of $\Bbb Z_p$.