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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?

azimut
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Haded1027
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  • Thanks for the update. Have you seen the 2014 paper by V. Sison, "Bases of the Galois Ring $GR(p^r,m)$ over the Integer Ring $\mathbb{Z}_{p^r}$"? The author constructs what he terms generalized Frobenius automorphisms of said rings. That study was also motivated by a code-theoretic problem. – hardmath Nov 26 '16 at 00:35
  • Yes I've read it and found that generalized Frobenius maps are indeed ring automorphisms. But I was wondering, are there more ring automorphisms other than that? Or finding all ring automorphisms of Galois rings are tedious and cannot be enumerated easily? – Haded1027 Nov 26 '16 at 04:09
  • I don't know if these generalized Frobenius automorphisms generate all of $Aut(GR(p^r,m)$. I'm not a specialist, but to my superficial understanding it seems a plausible conjecture. The book Finite commutative rings and their applications (Kluwer AP, 2002) by Bini and Flamini in Sec. 6.2 asserts (p. 105) the group of automorphisms of $GR(p^r,m)$ fixing $\mathbb{Z}{p^r}$ is isomorphic to the Galois group of $\mathbb{F}{p^m}$ over $\mathbb{F}_{p}$, citing developments earlier in the book for separable extensions of local rings. – hardmath Nov 26 '16 at 13:32

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

Raghavendran, R.: 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

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

azimut
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