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In mathematics, Galois rings are a type of finite commutative rings which generalize both the finite fields and the the rings of integers modulo a prime power. A Galois ring is constructed from the ring $\mathbb {Z} /p^{n}\mathbb {Z}$ similar to how a finite field $\mathbb {F} _{p^{r}}$ is constructed from $\mathbb {F} _{p}$ . It is a Galois extension of $\mathbb {Z} /p^{n}\mathbb {Z}$ , 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 $\mathbb {Z} /4\mathbb {Z}$ 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

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

where $f(x)\in \mathbb {Z} [x]$ 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 $\mathbb {Z} [x]/(4,x^{3}+2x^{2}+x-1)$ , or equivalently, $(\mathbb {Z} /4\mathbb {Z} )[\xi ]$ where $\xi$ 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 $(\mathbb {Z} /4\mathbb {Z} )[x]$ , which makes $\xi$ a 7th root of unity in GR(4, 3). The elements of GR(4, 3) can all be written in the form $a_{2}\xi ^{2}+a_{1}\xi +a_{0}$ where each of a₀, a₁, and a₂ is in $\mathbb {Z} /4\mathbb {Z}$ . For example, $\xi ^{3}=2\xi ^{2}-\xi +1$ and $\xi ^{4}=2\xi ^{3}-\xi ^{2}+\xi =-\xi ^{2}-\xi +2$ .^[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 $\mathbb {Z} [x]/(p^{n},f(x))$ when f is chosen to be a primitive polynomial. This means that, in $(\mathbb {Z} /p^{n}\mathbb {Z} )[x]$ , the polynomial $f(x)$ divides $x^{p^{r}-1}-1$ and does not divide $x^{m}-1$ 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 $\mathbb {F} _{p}$ and using Hensel lifting.^[9]

A primitive (p^r – 1)-th root of unity $\xi$ 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

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

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

Ideals, quotients, and subrings

Every Galois ring is a local ring. The unique maximal ideal is the principal ideal $(p)=p\operatorname {GR} (p^{n},r)$ , consisting of all elements which are multiples of p. The residue field $\operatorname {GR} (p^{n},r)/(p)$ is isomorphic to the finite field of order p^r. Furthermore, $(0),(p^{n-1}),...,(p),(1)$ are all the ideals.^[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 $G_{2}\cong (C_{p^{n-1}})^{r}$ , the direct product of r copies of the cyclic group of order pⁿ⁻¹
if p = 2 and n ≥ 3, then $G_{2}\cong C_{2}\times C_{2^{n-2}}\times (C_{2^{n-1}})^{r-1}$

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 $\mathbb {F} _{p^{r}}$ , 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

\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 $\alpha _{i}$ is in the set $\{0,1,\xi ,\xi ^{2},...,\xi ^{p^{r}-2}\}$ ) is an automorphism, which is called the generalized Frobenius automorphism. The fixed points of the generalized Frobenius automorphism are the elements of the subring $\mathbb {Z} /p^{n}\mathbb {Z}$ . 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 $\mathbb {Z} /p^{n}\mathbb {Z}$ , and the ring GR(pⁿ, r) is a Galois extension of $\mathbb {Z} /p^{n}\mathbb {Z}$ . 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 $\operatorname {Gal} (\mathbb {F} _{p^{r}}/\mathbb {F} _{p^{s}})$ .^[14]^[13]

References

^ Krull, Wolfgang (1924), "Algebraische Theorie der zerlegbaren Ringe (Algebraische Theorie der Ringe III)", Mathematische Annalen, 92: 183–213, JFM 50.0072.02
^ Janusz, G. J. (1966), "Separable algebras over commutative rings", Transactions of the American Mathematical Society, 122: 461–479, doi:10.2307/1994561, Zbl 0141.03402
^ Raghavendran 1969, p. 206
^ ^a ^b van Lint, J.H. (1999), Introduction to Coding Theory (3rd ed.), Springer, Chapter 8: Codes over ℤ₄, ISBN 978-3-540-64133-9
^ 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
^ ^a ^b McDonald 1974, p. 308
^ ^a ^b Bini & Flamini 2002, pp. 82–83
^ Raghavendran 1969, p. 207
^ ^a ^b Wan 2003, p. 316, Theorem 14.8
^ Bini & Flamini 2002, p. 95, Proposition 6.2.3
^ Wan 2003, p. 319, Theorem 14.11
^ Raghavendran 1969, p. 213
^ ^a ^b Wan 2013, pp. 327–331, Section 14.6
^ 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

[1] Krull, Wolfgang (1924), "Algebraische Theorie der zerlegbaren Ringe (Algebraische Theorie der Ringe III)", Mathematische Annalen, 92: 183–213, JFM 50.0072.02

[2] Janusz, G. J. (1966), "Separable algebras over commutative rings", Transactions of the American Mathematical Society, 122: 461–479, doi:10.2307/1994561, Zbl 0141.03402

[3] Raghavendran 1969, p. 206

[van-lint-4] van Lint, J.H. (1999), Introduction to Coding Theory (3rd ed.), Springer, Chapter 8: Codes over ℤ₄, 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

[mcdonald-p308-6] McDonald 1974, p. 308

[bini-flamini-82-83-7] Bini & Flamini 2002, pp. 82–83

[8] Raghavendran 1969, p. 207

[wan-theorem-14-8-9] 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

[wan-section-14-6-13] Wan 2013, pp. 327–331, Section 14.6

[14] Bini & Flamini 2002, p. 105

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+In mathematics, '''Galois rings''' are a type of [[finite ring|finite]] [[commutative ring]]s which generalize both the [[finite field]]s and the the [[ring of integers modulo n|rings of integers modulo]] a [[prime power]]. A Galois ring is constructed from the ring <math>\mathbb{Z}/p^n\mathbb{Z}</math> similar to how a finite field <math>\mathbb{F}_{p^r}</math> is constructed from <math>\mathbb{F}_p</math>. It is a [[Galois extension]] of <math>\mathbb{Z}/p^n\mathbb{Z}</math>, when the concept of a Galois extension is generalized beyond the context of [[Field (mathematics)|field]]s.
-#REDIRECT [[Finite ring]]
+Galois rings were studied by [[Wolfgang Krull|Krull]] (1924),<ref>{{citation |first=Wolfgang |last=Krull |author-link=Wolfgang Krull |title=Algebraische Theorie der zerlegbaren Ringe (Algebraische Theorie der Ringe III) |journal=Mathematische Annalen |volume=92 |pages=183-213 |date=1924 |url=https://gdz.sub.uni-goettingen.de/id/PPN235181684_0092?tify=%7B%22view%22:%22info%22,%22pages%22:%5B187%5D%7D |jfm=50.0072.02}}</ref> and independently by Janusz (1966)<ref>{{citation |first=G. J. |last=Janusz |title= Separable algebras over commutative rings |journal=Transactions of the American Mathematical Society |volume=122 |pages=461-479 |date=1966 |doi=10.2307/1994561 |zbl=0141.03402}}</ref> and by Raghavendran (1969),<ref>{{Harvnb|Raghavendran|1969|p=206}}</ref> 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 code]]s over <math>\Z / 4\Z</math> using Galois rings GR(4, ''r'').<ref name="van-lint">{{citation |first=J.H. |last=van Lint |title=Introduction to Coding Theory |url=https://archive.org/details/introductiontoco0000lint_a3b9 |url-access=registration |date=1999 |publisher=Springer |isbn=978-3-540-64133-9 |edition=3rd |at=Chapter 8: Codes over ℤ<sub>4</sub>}}</ref><ref>{{citation |first1=A.R. |last1=Hammons |first2=P.V. |last2=Kumar |first3=A.R. |last3=Calderbank |first4=N.J.A. |last4=Sloane |first5=P. |last5=Solé| title=The Z_4-linearity of Kerdock, Preparata, Goethals, and related codes |journal=IEEE Transactions on Information Theory |volume=40 |date=1994 |pp=301–319 |url=http://neilsloane.com/doc/Me184.pdf}}</ref>
+==Definition==
+A Galois ring is a commutative ring of [[Characteristic (algebra)|characteristic]] ''p''<sup>''n''</sup> which has ''p''<sup>''nr''</sup> elements, where ''p'' is prime and ''n'' and ''r'' are positive integers. It is usually denoted GR(''p''<sup>''n''</sup>, ''r''). It can be defined as a [[quotient ring]]
+:<math>\operatorname{GR}(p^n, r) \cong \mathbb{Z}[x]/(p^n, f(x))</math>
+where <math>f(x) \in \Z[x]</math> is a monic polynomial of degree ''r'' which is [[Irreducible polynomial|irreducible]] modulo ''p''.<ref name="mcdonald-p308">{{harvnb|McDonald|1974|p=308}}</ref><ref name="bini-flamini-82-83">{{harvnb|Bini|Flamini|2002|pp=82–83}}</ref> Up to isomorphism, the ring depends only on ''p'', ''n'', and ''r'' and not on the choice of ''f'' used in the construction.<ref>{{harvnb|Raghavendran|1969|p=207}}</ref>
+==Examples==
+The simplest examples of Galois rings are important special cases:
+* The Galois ring GR(''p''<sup>''n''</sup>, 1) is the [[ring of integers modulo n|ring of integers modulo ''p''<sup>''n''</sup>]].
+* The Galois ring GR(''p'', ''r'') is the [[finite field]] of order ''p''<sup>''r''</sup>.
+A less trivial example is the Galois ring GR(4, 3). It is of characteristic 4 and has 4<sup>3</sup> = 64 elements. One way to construct it is <math>\mathbb{Z}[x]/(4, x^3 + 2x^2 + x - 1)</math>, or equivalently, <math>(\mathbb{Z}/4\mathbb{Z})[\xi]</math> where <math>\xi</math> is a root of the polynomial <math>f(x) = x^3 + 2x^2 + x - 1</math>. 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
+:<math>x^7 - 1 = (x^3 + 2x^2 + x - 1)(x^3 - x^2 + 2x - 1)(x - 1)</math>
+in <math>(\mathbb{Z}/4\mathbb{Z})[x]</math>, which makes <math>\xi</math> a 7th [[root of unity]] in GR(4, 3). The elements of GR(4, 3) can all be written in the form <math>a_2 \xi^2 + a_1 \xi + a_0</math> where each of ''a''<sub>0</sub>, ''a''<sub>1</sub>, and ''a''<sub>2</sub> is in <math>\mathbb{Z}/4\mathbb{Z}</math>. For example, <math>\xi^3 = 2\xi^2 - \xi + 1</math> and <math>\xi^4 = 2\xi^3 - \xi^2 + \xi = -\xi^2 - \xi + 2</math>.<ref name="van-lint" />
+==Structure==
+===(''p''<sup>''r''</sup> – 1)-th roots of unity===
+Every Galois ring GR(''p''<sup>''n''</sup>, ''r'') has a [[Primitive root of unity|primitive ({{nowrap|''p''<sup>''r''</sup> – 1}})-th root of unity]]. It is the equivalence class of ''x'' in the quotient <math>\mathbb{Z}[x]/(p^n, f(x))</math> when ''f'' is chosen to be a ''primitive polynomial''. This means that, in <math>(\mathbb{Z}/p^n\mathbb{Z})[x]</math>, the polynomial <math>f(x)</math> divides <math>x^{p^r - 1} - 1</math> and does not divide <math>x^m - 1</math> for all {{nowrap|''m'' < ''p''<sup>''r''</sup> – 1}}. Such an ''f'' can be computed by starting with a [[Primitive polynomial (field theory)|primitive polynomial]] of degree ''r'' over the finite field <math>\mathbb{F}_p</math> and using [[Hensel lifting]].<ref name="wan-theorem-14-8" />
+A primitive ({{nowrap|''p''<sup>''r''</sup> – 1}})-th root of unity <math>\xi</math> 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
+:<math>\alpha_0 + \alpha_1 p + \cdots + \alpha_{n-1} p^{n-1}</math>
+where each <math>\alpha_i</math> is in the set <math>\{0, 1, \xi, \xi^2, ..., \xi^{p^r - 2}\}</math>.<ref name="bini-flamini-82-83" /><ref name="wan-theorem-14-8">{{harvnb|Wan|2003|p=316|loc=Theorem 14.8}}</ref>
+===Ideals, quotients, and subrings===
+Every Galois ring is a [[local ring]]. The unique [[maximal ideal]] is the [[principal ideal]] <math>(p) = p \operatorname{GR}(p^n, r)</math>, consisting of all elements which are multiples of ''p''. The [[residue field]] <math>\operatorname{GR}(p^n, r)/(p)</math> is isomorphic to the finite field of order ''p''<sup>''r''</sup>. Furthermore, <math>(0), (p^{n-1}), ..., (p), (1)</math> are all the ideals.<ref name="mcdonald-p308" />
+The Galois ring GR(''p''<sup>''n''</sup>, ''r'') contains a unique [[subring]] isomorphic to GR(''p''<sup>''n''</sup>, ''s'') for every ''s'' which divides ''r''. These are the only subrings of GR(''p''<sup>''n''</sup>, ''r'').<ref>{{harvnb|Bini|Flamini|2002|p=95|loc=Proposition 6.2.3}}</ref>
+===Group of units===
+The [[Unit (ring theory)|units]] of a Galois ring ''R'' are all the elements which are not multiples of ''p''. The group of units, ''R''<sup>×</sup>, can be decomposed as a [[Direct product of groups|direct product]] ''G''<sub>1</sub>×''G''<sub>2</sub>, as follows. The subgroup ''G''<sub>1</sub> is the group of ({{nowrap|''p''<sup>''r''</sup> – 1}})-th roots of unity. It is a [[cyclic group]] of order {{nowrap|''p''<sup>''r''</sup> – 1}}. The subgroup ''G''<sub>2</sub> is 1+''pR'', consisting of all elements congruent to 1 modulo ''p''. It is a group of order ''p''<sup>''r''(''n''−1)</sup>, with the following structure:
+* if ''p'' is odd or if ''p'' = 2 and ''n'' ≤ 2, then <math>G_2 \cong (C_{p^{n-1}})^r</math>, the direct product of ''r'' copies of the cyclic group of order ''p''<sup>''n''−1</sup>
+* if ''p'' = 2 and ''n'' ≥ 3, then <math>G_2 \cong C_2 \times C_{2^{n-2}} \times (C_{2^{n-1}})^{r-1}</math>
+This description generalizes the structure of the [[Multiplicative group of integers modulo n|multiplicative group of integers modulo ''p''<sup>''n''</sup>]], which is the case ''r'' = 1.<ref>{{harvnb|Wan|2003|p=319|loc=Theorem 14.11}}</ref>
+===Automorphisms===
+Analogous to the automorphisms of the finite field <math>\mathbb{F}_{p^r}</math>, the [[automorphism group]] of the Galois ring GR(''p''<sup>''n''</sup>, ''r'') is a cyclic group of order ''r''.<ref>{{Harvnb|Raghavendran|1969|p=213}}</ref> The automorphisms can be described explicitly using the ''p''-adic representation. Specifically, the map
+:<math>\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}</math>
+(where each <math>\alpha_i</math> is in the set <math>\{0, 1, \xi, \xi^2, ..., \xi^{p^r - 2}\}</math>) is an automorphism, which is called the generalized [[Frobenius automorphism]]. The [[Fixed point (mathematics)|fixed point]]s of the generalized Frobenius automorphism are the elements of the subring <math>\mathbb{Z}/p^n\mathbb{Z}</math>. Iterating the generalized Frobenius automorphism gives all the automorphisms of the Galois ring.<ref name="wan-section-14-6">{{harvnb|Wan|2013|pp=327–331|loc=Section 14.6}}</ref>
+The automorphism group can be thought of as the [[Galois group]] of GR(''p''<sup>''n''</sup>, ''r'') over <math>\mathbb{Z}/p^n\mathbb{Z}</math>, and the ring GR(''p''<sup>''n''</sup>, ''r'') is a [[Galois extension]] of <math>\mathbb{Z}/p^n\mathbb{Z}</math>. More generally, whenever ''r'' is a multiple of ''s'', GR(''p''<sup>''n''</sup>, ''r'') is a Galois extension of GR(''p''<sup>''n''</sup>, ''s''), with Galois group isomorphic to <math>\operatorname{Gal}(\mathbb{F}_{p^r} / \mathbb{F}_{p^s})</math>.<ref>{{harvnb|Bini|Flamini|2002|p=105}}</ref><ref name="wan-section-14-6" />
+==References==
+{{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]]