Questions tagged [galois-rings]

Galois rings are a class of finite commutative rings generalizing both the finite fields and the integer residue rings modulo a prime power in a quite natural way. Their name stems from the fact that they share a lot of properties with the finite fields.

A Galois ring $R$ is uniquely determined by its order and its characteristic. The characteristic is always a prime power $p^m$ with $p$ prime, and the order is $p^{mr}$ with a positive integer $r$. In this situation, $R$ is denoted by $\operatorname{GR}(p^m,r)$ and $r$ is called the rank of $R$. We have $\operatorname{GR}(p^1,r) \cong \mathbb{F}_{p^r}$ and $\operatorname{GR}(p^m,1) \cong \mathbb{Z}/p^m\mathbb{Z}$.

The ring $R$ has the ideals $p^i R$ with $i\in\{0,\ldots,m\}$. So the lattice of ideals is a chain, and the unique maximal ideal is $p^{m-1} R$. The factor ring $R/ p^{m-1} R$ is isomorphic to $\mathbb{F}_{p^r}$ and called the residue field of $R$.

There are three common ways to define the Galois ring $\operatorname{GR}(p^m,r)$.

1. Bottom-up: Let $f\in\mathbb{Z}/(p^m)[X]$ be a monic polynomial of degree $r$ whose image modulo $p$ is irreducible in $\mathbb{F}_p[X]$. Then $$\operatorname{GR}(p^m,r) := \mathbb{Z}/(p^m)[X]/(f)\text{.}$$ Up to isomorphism, this definition does not depend on the exact choice of $f$.

2. Top-down: Let $\zeta$ be a primitive $(p^r - 1)$-st root of unity over $\mathbb{Q}_p$. The ring of integers of $\mathbb{Q}_p[\zeta]$ is given by $\mathbb{Z}_p[\zeta]$. Now $$\operatorname{GR}(p^m,r) := \mathbb{Z}_p[\zeta]/(p^m)\text{.}$$

3. Let $W(\mathbb{F}_{p^r})$ be the ring of Witt vectors over $\mathbb{F}_{p^r}$. Then $\operatorname{GR}(p^m,r)$ is the factor ring of $W(\mathbb{F}_{p^r})$ arising by truncation after the $m$-th position.