Galois ring

The simplest examples of Galois rings are important special cases:

• The Galois ring GR(pⁿ, 1) is the ring of integers modulo pⁿ.
• The Galois ring GR(p, r) is the finite field of order p^r.

A less trivial example is the Galois ring GR(4, 3). It is of characteristic 4 and has 4³ = 64 elements. One way to construct it is ${\displaystyle \mathbb {Z} [x]/(4,x^{3}+2x^{2}+x-1)}$, or equivalently, ${\displaystyle (\mathbb {Z} /4\mathbb {Z} )[\xi ]}$ where ${\displaystyle \xi }$ is a root of the polynomial ${\displaystyle f(x)=x^{3}+2x^{2}+x-1}$. Although any monic polynomial of degree 3 which is irreducible modulo 2 could have been used, this choice of f turns out to be convenient because

${\displaystyle x^{7}-1=(x^{3}+2x^{2}+x-1)(x^{3}-x^{2}+2x-1)(x-1)}$

in ${\displaystyle (\mathbb {Z} /4\mathbb {Z} )[x]}$, which makes ${\displaystyle \xi }$ a 7th root of unity in GR(4, 3). The elements of GR(4, 3) can all be written in the form ${\displaystyle a_{2}\xi ^{2}+a_{1}\xi +a_{0}}$ where each of a₀, a₁, and a₂ is in ${\displaystyle \mathbb {Z} /4\mathbb {Z} }$. For example, ${\displaystyle \xi ^{3}=2\xi ^{2}-\xi +1}$ and ${\displaystyle \xi ^{4}=2\xi ^{3}-\xi ^{2}+\xi =-\xi ^{2}-\xi +2}$.^[4]

Every Galois ring GR(pⁿ, r) has a primitive (p^r – 1)-th root of unity. It is the equivalence class of x in the quotient ${\displaystyle \mathbb {Z} [x]/(p^{n},f(x))}$ when f is chosen to be a primitive polynomial. This means that, in ${\displaystyle (\mathbb {Z} /p^{n}\mathbb {Z} )[x]}$, the polynomial ${\displaystyle f(x)}$ divides ${\displaystyle x^{p^{r}-1}-1}$ and does not divide ${\displaystyle x^{m}-1}$ for all m < p^r – 1. Such an f can be computed by starting with a primitive polynomial of degree r over the finite field ${\displaystyle \mathbb {F} _{p}}$ and using Hensel lifting.^[9]

A primitive (p^r – 1)-th root of unity ${\displaystyle \xi }$ can be used to express elements of the Galois ring in a useful form called the p-adic representation. Every element of the Galois ring can be written uniquely as

${\displaystyle \alpha _{0}+\alpha _{1}p+\cdots +\alpha _{n-1}p^{n-1}}$

where each ${\displaystyle \alpha _{i}}$ is in the set ${\displaystyle \{0,1,\xi ,\xi ^{2},...,\xi ^{p^{r}-2}\}}$.^[7][9]

Every Galois ring is a local ring. The unique maximal ideal is the principal ideal ${\displaystyle (p)=p\operatorname {GR} (p^{n},r)}$, consisting of all elements which are multiples of p. The residue field ${\displaystyle \operatorname {GR} (p^{n},r)/(p)}$ is isomorphic to the finite field of order p^r. Furthermore, ${\displaystyle (0),(p^{n-1}),...,(p),(1)}$ are all the ideals.^[6]

The Galois ring GR(pⁿ, r) contains a unique subring isomorphic to GR(pⁿ, s) for every s which divides r. These are the only subrings of GR(pⁿ, r).^[10]

The units of a Galois ring R are all the elements which are not multiples of p. The group of units, R^×, can be decomposed as a direct product G₁×G₂, as follows. The subgroup G₁ is the group of (p^r − 1)-th roots of unity. It is a cyclic group of order p^r − 1. The subgroup G₂ is 1+pR, consisting of all elements congruent to 1 modulo p. It is a group of order p^r(n−1), with the following structure:

• if p is odd or if p = 2 and n ≤ 2, then ${\displaystyle G_{2}\cong (C_{p^{n-1}})^{r}}$, the direct product of r copies of the cyclic group of order pⁿ⁻¹
• if p = 2 and n ≥ 3, then ${\displaystyle G_{2}\cong C_{2}\times C_{2^{n-2}}\times (C_{2^{n-1}})^{r-1}}$

This description generalizes the structure of the multiplicative group of integers modulo pⁿ, which is the case r = 1.^[11]

Analogous to the automorphisms of the finite field ${\displaystyle \mathbb {F} _{p^{r}}}$, the automorphism group of the Galois ring GR(pⁿ, r) is a cyclic group of order r.^[12] The automorphisms can be described explicitly using the p-adic representation. Specifically, the map

${\displaystyle \phi (\alpha _{0}+\alpha _{1}p+\cdots +\alpha _{n-1}p^{n-1})=\alpha _{0}^{p}+\alpha _{1}^{p}p+\cdots +\alpha _{n-1}^{p}p^{n-1}}$

(where each ${\displaystyle \alpha _{i}}$ is in the set ${\displaystyle \{0,1,\xi ,\xi ^{2},...,\xi ^{p^{r}-2}\}}$) is an automorphism, which is called the generalized Frobenius automorphism. The fixed points of the generalized Frobenius automorphism are the elements of the subring ${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }$. Iterating the generalized Frobenius automorphism gives all the automorphisms of the Galois ring.^[13]

The automorphism group can be thought of as the Galois group of GR(pⁿ, r) over ${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }$, and the ring GR(pⁿ, r) is a Galois extension of ${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }$. More generally, whenever r is a multiple of s, GR(pⁿ, r) is a Galois extension of GR(pⁿ, s), with Galois group isomorphic to ${\displaystyle \operatorname {Gal} (\mathbb {F} _{p^{r}}/\mathbb {F} _{p^{s}})}$.^[14][13]

• McDonald, Bernard A. (1974), Finite Rings with Identity, Marcel Dekker, ISBN 978-0-8247-6161-5, Zbl 0294.16012
• Bini, G; Flamini, F (2002), Finite commutative rings and their applications, Kluwer, ISBN 978-1-4020-7039-6, Zbl 1095.13032
• Raghavendran, R. (1969), "Finite associative rings", Compositio Mathematica, 21 (2): 195–229, Zbl 0179.33602
• Wan, Zhe-Xian (2003), Lectures on finite fields and Galois rings, World Scientific, ISBN 981-238-504-5, Zbl 1028.11072