Galois ring

A Galois ring is a commutative ring of characteristic pⁿ which has p^nr elements, where p is prime and n and r are positive integers. It is usually denoted GR(pⁿ, r). It can be defined as a quotient ring

:$\backslash operatorname\{GR\}(p^n,\; r)\; \backslash cong\; \backslash mathbb\{Z\}[x]/(p^n,\; f(x))$

where $f(x)\; \backslash in\; \backslash Z[x]$ is a monic polynomial of degree r which is irreducible modulo p.{{harvnb|McDonald|1974|p=308}}{{harvnb|Bini|Flamini|2002|pp=82–83}} Up to isomorphism, the ring depends only on p, n, and r and not on the choice of f used in the construction.{{harvnb|Raghavendran|1969|p=207}}

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 $\backslash mathbb\{Z\}[x]/(4,\; x^3\; +\; 2x^2\; +\; x\; -\; 1)$, or equivalently, $(\backslash mathbb\{Z\}/4\backslash mathbb\{Z\})[\backslash xi]$ where $\backslash xi$ is a root of the polynomial $f(x)\; =\; x^3\; +\; 2x^2\; +\; x\; -\; 1$. 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

:$x^7\; -\; 1\; =\; (x^3\; +\; 2x^2\; +\; x\; -\; 1)(x^3\; -\; x^2\; +\; 2x\; -\; 1)(x\; -\; 1)$

in $(\backslash mathbb\{Z\}/4\backslash mathbb\{Z\})[x]$, which makes $\backslash xi$ a 7th root of unity in GR(4, 3). The elements of GR(4, 3) can all be written in the form $a\_2\; \backslash xi^2\; +\; a\_1\; \backslash xi\; +\; a\_0$ where each of a₀, a₁, and a₂ is in $\backslash mathbb\{Z\}/4\backslash mathbb\{Z\}$. For example, $\backslash xi^3\; =\; 2\backslash xi^2\; -\; \backslash xi\; +\; 1$ and $\backslash xi^4\; =\; 2\backslash xi^3\; -\; \backslash xi^2\; +\; \backslash xi\; =\; -\backslash xi^2\; -\; \backslash xi\; +\; 2$.

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

A primitive ({{nowrap|p^r – 1}})-th root of unity $\backslash xi$ 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

:$\backslash alpha\_0\; +\; \backslash alpha\_1\; p\; +\; \backslash cdots\; +\; \backslash alpha\_\{n-1\}\; p^\{n-1\}$

where each $\backslash alpha\_i$ is in the set $\backslash \{0,\; 1,\; \backslash xi,\; \backslash xi^2,\; ...,\; \backslash xi^\{p^r\; -\; 2\}\backslash \}$.{{harvnb|Wan|2003|p=316|loc=Theorem 14.8}}

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

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).{{harvnb|Bini|Flamini|2002|p=95|loc=Proposition 6.2.3}}

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 ({{nowrap|p^r − 1}})-th roots of unity. It is a cyclic group of order {{nowrap|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 $G\_2\; \backslash cong\; (C\_\{p^\{n-1\}\})^r$, the direct product of r copies of the cyclic group of order pⁿ⁻¹
• if p = 2 and n ≥ 3, then $G\_2\; \backslash cong\; C\_2\; \backslash times\; C\_\{2^\{n-2\}\}\; \backslash times\; (C\_\{2^\{n-1\}\})^\{r-1\}$

This description generalizes the structure of the multiplicative group of integers modulo pⁿ, which is the case r = 1.{{harvnb|Wan|2003|p=319|loc=Theorem 14.11}}

Analogous to the automorphisms of the finite field $\backslash mathbb\{F\}\_\{p^r\}$, the automorphism group of the Galois ring GR(pⁿ, r) is a cyclic group of order r.{{Harvnb|Raghavendran|1969|p=213}} The automorphisms can be described explicitly using the p-adic representation. Specifically, the map

:$\backslash phi(\backslash alpha\_0\; +\; \backslash alpha\_1\; p\; +\; \backslash cdots\; +\; \backslash alpha\_\{n-1\}\; p^\{n-1\})\; =\; \backslash alpha\_0^p\; +\; \backslash alpha\_1^p\; p\; +\; \backslash cdots\; +\; \backslash alpha\_\{n-1\}^p\; p^\{n-1\}$

(where each $\backslash alpha\_i$ is in the set $\backslash \{0,\; 1,\; \backslash xi,\; \backslash xi^2,\; ...,\; \backslash xi^\{p^r\; -\; 2\}\backslash \}$) is an automorphism, which is called the generalized Frobenius automorphism. The fixed points of the generalized Frobenius automorphism are the elements of the subring $\backslash mathbb\{Z\}/p^n\backslash mathbb\{Z\}$. Iterating the generalized Frobenius automorphism gives all the automorphisms of the Galois ring.{{harvnb|Wan|2003|pp=327–331|loc=Section 14.6}}

The automorphism group can be thought of as the Galois group of GR(pⁿ, r) over $\backslash mathbb\{Z\}/p^n\backslash mathbb\{Z\}$, and the ring GR(pⁿ, r) is a Galois extension of $\backslash mathbb\{Z\}/p^n\backslash mathbb\{Z\}$. 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 $\backslash operatorname\{Gal\}(\backslash mathbb\{F\}\_\{p^r\}\; /\; \backslash mathbb\{F\}\_\{p^s\})$.{{harvnb|Bini|Flamini|2002|p=105}}

{{reflist}}

• {{citation |first=Bernard A. |last=McDonald |title=Finite Rings with Identity |publisher=Marcel Dekker |date=1974 |isbn=978-0-8247-6161-5 |zbl=0294.16012}}
• {{citation |first1=G |last1=Bini |first2=F |last2=Flamini |title=Finite commutative rings and their applications |publisher=Kluwer |year=2002 |isbn=978-1-4020-7039-6 |zbl=1095.13032 |url=https://books.google.com/books?id=H0mAM-Zr0HAC}}
• {{citation |first=R. |last=Raghavendran |title=Finite associative rings |journal=Compositio Mathematica |volume=21 |issue=2 |pages=195–229 |year=1969 |url=http://www.numdam.org/item?id=CM_1969__21_2_195_0 |zbl=0179.33602}}
• {{citation |last=Wan |first=Zhe-Xian |title=Lectures on finite fields and Galois rings |year=2003 |publisher=World Scientific |isbn=981-238-504-5 |url=https://books.google.com/books?id=F_A7DQAAQBAJ |zbl=1028.11072}}

Category:Finite rings