Galois ring

In mathematics, Galois rings are a type of finite commutative rings which generalize both the finite fields and the rings of integers modulo a prime power. A Galois ring is constructed from the ring ${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }$ similar to how a finite field ${\displaystyle \mathbb {F} _{p^{r}}}$ is constructed from ${\displaystyle \mathbb {F} _{p}}$. It is a Galois extension of ${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }$, when the concept of a Galois extension is generalized beyond the context of fields.

Galois rings were studied by Krull (1924),^[1] and independently by Janusz (1966)^[2] and by Raghavendran (1969),^[3] who both introduced the name Galois ring. They are named after Évariste Galois, similar to Galois fields, which is another name for finite fields. Galois rings have found applications in coding theory, where certain codes are best understood as linear codes over ${\displaystyle \mathbb {Z} /4\mathbb {Z} }$ using Galois rings GR(4, r).^[4][5]

Definition

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

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

where ${\displaystyle f(x)\in \mathbb {Z} [x]}$ is a monic polynomial of degree r which is irreducible modulo p.^[6][7] Up to isomorphism, the ring depends only on p, n, and r and not on the choice of f used in the construction.^[8]

Examples

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]

Structure

(p^r – 1)-th roots of unity

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]

Ideals, quotients, and subrings

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]

Group of units

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]

Automorphisms

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]

References

1. Krull, Wolfgang (1924), "Algebraische Theorie der zerlegbaren Ringe (Algebraische Theorie der Ringe III)", Mathematische Annalen, 92: 183–213, doi:10.1007/BF01448006, JFM 50.0072.02, S2CID 116728217
2. Janusz, G. J. (1966), "Separable algebras over commutative rings", Transactions of the American Mathematical Society, 122 (2): 461–479, doi:10.2307/1994561, JSTOR 1994561, Zbl 0141.03402
3. Raghavendran 1969, p. 206
4. van Lint, J.H. (1999), Introduction to Coding Theory (3rd ed.), Springer, Chapter 8: Codes over ${\displaystyle \mathbb {Z} }$₄, ISBN 978-3-540-64133-9
5. Hammons, A.R.; Kumar, P.V.; Calderbank, A.R.; Sloane, N.J.A.; Solé, P. (1994), "The Z_4-linearity of Kerdock, Preparata, Goethals, and related codes" (PDF), IEEE Transactions on Information Theory, 40: 301–319, doi:10.1109/18.312154, S2CID 7667081
6. McDonald 1974, p. 308
7. Bini & Flamini 2002, pp. 82–83
8. Raghavendran 1969, p. 207
9. Wan 2003, p. 316, Theorem 14.8
10. Bini & Flamini 2002, p. 95, Proposition 6.2.3
11. Wan 2003, p. 319, Theorem 14.11
12. Raghavendran 1969, p. 213
13. Wan 2003, pp. 327–331, Section 14.6
14. Bini & Flamini 2002, p. 105

• 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