# Galois ring

The Galois ring ${ \mathop{\rm GR} } ( p ^ {m} ,d )$ is [a5] the unique Galois extension of $\mathbf Z/ {p ^ {m} } \mathbf Z$ of degree $d$. For instance ${ \mathop{\rm GR} } ( p ^ {m} ,1 )$ is $\mathbf Z/p ^ {m} \mathbf Z$ and ${ \mathop{\rm GR} } ( p,d )$ is $\mathbf F _ {p ^ {d} }$. Generalizing finite fields (cf. Finite field), these rings find applications in similar areas: linear recurrences [a1], [a6], [a7], cyclic codes [a9], [a2] [a3], association schemes [a10], and character sums [a10], [a4]. For a connection with Witt rings see [a8] (cf. also Witt ring). Two different constructions of these rings are given below: bottom-up, starting from a finite field, and top-down, starting from a local field.

## Bottom up.

This is the first and the most algorithmic one. Let $n = p ^ {d} - 1$. Pick an irreducible monic primitive polynomial ${\overline{f}\; }$ of degree $d$, as in the standard construction of $\mathbf F _ {p ^ {d} }$ from $\mathbf F _ {p}$, and lift it to a polynomial $f$ over $\mathbf Z/p ^ {m} \mathbf Z$ in such a way that the nice finite field property ( $f$ divides $X ^ {n} - 1$) still holds. In the language of linear recurrences (or linear feedback shift registers), one has lifted an $m$- sequence of period $n$ over $\mathbf F _ {p}$ into a linear recurrence over $\mathbf Z/p ^ {m} \mathbf Z$ of the same period. This is construction $A$ of [a1]. Note that an arbitrary lift will lead to multiplying the period by a power of $p$, as in construction $B$ of [a1]. For example, $p = 2$, $f ( x ) = x ^ {2} + x + 1$ gives a period $6$ and not $3$. Now, let

$${ \mathop{\rm GR} } ( p ^ {m} ,d ) = \mathbf Z/p ^ {m} \mathbf Z [ X ] / ( f ) .$$

## Top down.

This $p$- adic approach was introduced in print in [a4] but was implicitly known to M. Yamada [a10], who used the term "Teichmüller" , as in $p$- adic analysis, and also to E. Spiegel [a9]. Denote by $\mathbf Z _ {p}$ the ring of $p$- adic integers, best viewed as the set of formal expansions in powers of $p$ with coefficients in the residue field $\mathbf F _ {p}$. Then ${ \mathop{\rm GR} } ( p ^ {m} ,1 ) = \mathbf Z _ {p} / ( p ^ {m} )$. For higher values of $d$ one considers the unramified extension of $\mathbf Q _ {p}$ generated by $\zeta _ {n}$( an $n$- th root of unity) and its ring of integers $\mathbf Z _ {p} [ \zeta _ {n} ]$. Let $T _ {d}$ denote the set of $p ^ {d}$ roots of $1$ over this latter ring. This set of so-called Teichmüller representatives reduces modulo $p$ to $\mathbf F _ {p ^ {d} }$. The ring of integers of the $p$- adic field admits the following expansion: $\mathbf Z _ {p} [ \zeta _ {n} ] = \sum _ {i = 0 } ^ \infty p ^ {i} T ^ {i}$, which converges in the sense of the $p$- adic valuation. Modulo $p ^ {m}$ this yields

$${ \mathop{\rm GR} } ( p ^ {m} ,d ) = \mathbf Z _ {p} [ \zeta _ {n} ] / ( p ^ {m} ) .$$

## Multiplicative structure.

The ring $R = { \mathop{\rm GR} } ( p ^ {m} ,d )$ comprises units $R ^ \times$ and zero divisors $pR$. The multiplicative group $R ^ \times$ is the direct product of $T _ {d} \setminus \{ 0 \}$ by the group of so-called principal units $1 + pR$. The group of principal units is isomorphic, for $m = 2$ or $p > 2$, to the additive group of $( \mathbf Z/p ^ {m - 1 } \mathbf Z ) ^ {d}$. The Galois group of $R$ over $\mathbf Z/p ^ {m} \mathbf Z$ is isomorphic to the Galois group of $\mathbf F _ {p ^ {d} }$ over $\mathbf F _ {p}$ and therefore cyclic of order $d$.

How to Cite This Entry:
Galois ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Galois_ring&oldid=47033
This article was adapted from an original article by P. SolÃ© (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article