Galois ring

Builds on Integers modulo n Monic polynomial Irreducible polynomial Quotient ring Finite field

Definition

Let $p$ be a prime number and let $n \ge 1$ and $r \ge 1$ be integers. Let $f \in (\Z/p^n\Z)[x]$ be a monic polynomial of degree $r$ whose reduction modulo $p$ is irreducible in $\F_p[x]$ . The Galois ring of characteristic $p^n$ and rank $r$ is the quotient ring $\mathrm{GR}(p^n, r) = (\Z/p^n\Z)[x] / (f).$ Here $\Z/p^n\Z$ is the ring of integers modulo $p^n$ . Up to isomorphism, this ring depends only on $p$ , $n$ and $r$ , and not on the choice of $f$ .

Remarks

Take $p = 2$ , $n = 2$ , $r = 2$ and $f = x^2 + x + 1$ . The polynomial $f$ is monic, and modulo $2$ it has no root in $\F_2$ , so it is irreducible of degree $2$ over $\F_2$ . Hence $\mathrm{GR}(4, 2) = (\Z/4\Z)[x]/(x^2 + x + 1)$ is a Galois ring with $4^2 = 16$ elements. The ring $(\Z/4\Z)[x]/(x^2 + 1)$ is not obtained this way, because $x^2 + 1 \equiv (x + 1)^2$ modulo $2$ is reducible. The ring $\mathrm{GR}(p^n, r)$ has $p^{nr}$ elements and characteristic $p^n$ . It is a local ring with maximal ideal $p \, \mathrm{GR}(p^n, r)$ and residue field the finite field $\F_{p^r}$ . The special cases are $\mathrm{GR}(p, r) = \F_{p^r}$ and $\mathrm{GR}(p^n, 1) = \Z/p^n\Z$ .

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Finite field concept
Integers modulo n concept
Irreducible polynomial concept
Monic polynomial concept
Quotient ring concept
Galois ring definition

Tag	`8Y0M6X8Q2A`
URI	`m2://8Y0M6X8Q2A`
Kind	definition
Writing	W0 stub
MSC	13M05, 11T30
Defines	Galois ring
Uses	Integers modulo n, Monic polynomial, Irreducible polynomial, Quotient ring, Finite field
Changed	2 November 2026

Galois ring

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here