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.
similar to how a finite field
, 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.
[4][5] A Galois ring is a commutative ring of characteristic pn which has pnr elements, where p is prime and n and r are positive integers.
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] The simplest examples of Galois rings are important special cases: A less trivial example is the Galois ring GR(4, 3).
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 in
a 7th root of unity in GR(4, 3).
The elements of GR(4, 3) can all be written in the form
[4] Every Galois ring GR(pn, r) has a primitive (pr – 1)-th root of unity.
It is the equivalence class of x in the quotient
when f is chosen to be a primitive polynomial.
Such an f can be computed by starting with a primitive polynomial of degree r over the finite field
[9] A primitive (pr – 1)-th root of unity
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 where each
, consisting of all elements which are multiples of p. The residue field
is isomorphic to the finite field of order pr.
[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 (pr – 1)-th roots of unity.
It is a cyclic group of order pr – 1.
The subgroup G2 is 1+pR, consisting of all elements congruent to 1 modulo p. It is a group of order pr(n−1), with the following structure: 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
, 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.
The fixed points of the generalized Frobenius automorphism are the elements of the subring
, and the ring GR(pn, r) is a Galois extension of
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