Ring Automorphisms of Galois Rings

Question

I would like to know all ring automorphisms of $GR(p^r,m)$. Is there a way to do this? I am studying skew cyclic codes over Galois rings. However, ring automorphisms play a huge role in this. I am wondering whether or not the ring automorphisms of Galois rings are only the identity automorphism. If not, how can I find the non-identity automorphisms of Galois rings?

Answer 1

The answer turns out to be a pretty generalization of the well-known automorphism group of finite fields. It found in the article

Finite associative rings, Compositio Mathematica 21.2 (1969), 195–229.

The group of ring automorphisms of $R = \operatorname{GR}(p^r,m)$ is cyclic of order $m$. It is generated by the generalized Frobenius map.

To describe this map, we first define the set of Teichmüller units $T$ as the unique subgroup of $R^*$ of order $p^m-1$. One shows that each $r\in R$ has a unique $p$-adic expansion $r = \sum_{i=0}^{r-1} t_i p^i$ with $t_i\in T\cup\{0\}$. Now in $p$-adic expansion, the generalized Frobenius map has the form

$$F : \sum_{i=0}^{r-1} t_i p^i \mapsto \sum_{i=0}^{r-1} t_i^p p^i.$$