Introduction

Magma provides facilities for computing with Galois rings. The features are currently very basic, but advanced features will be available in the near future, including support for the creation of subrings and appropriate embeddings, allowing lattices of compatible embeddings, just as for finite fields. A Galois ring \(R\) in Magma is considered as a finite algebraic extension of \({\mathbb{Z}}_{p^a}\) (where \(p\) is prime) by a monic polynomial \(D\in{\mathbb{Z}}[x]\) which is irreducible modulo \(p\). Thus \(R\) is presented as the polynomial quotient ring \({\mathbb{Z}}_{p^a}[x]/\langle D\rangle\) and is usually written as \(\operatorname{GR}({p^a}, { d})\), where \(d\) is the degree of \(D\). The cardinality of \(R\) is easily seen to be \(p^{ad}\). \(R\) has a unique maximal ideal generated by \(p\), and the quotient ring \(R/\langle p\rangle\) is a finite field isomorphic to \({\mathbb{Z}}_p[x]/\langle D\rangle\), where \(D\) is here considered as a polynomial in \({\mathbb{Z}}_p[x]\) (the coefficients are reduced modulo \(p\)). This finite field is called the residue field of \(R\). In the following, we will also call the integer residue ring \({\mathbb{Z}}_{p^a}\) the base ring of \(R\), because this is the subring of \(R\) generated by 1 and we can think of \(R\) as an extension of \({\mathbb{Z}}_{p^a}\). For a non-zero element \(x\) of \(R\), the valuation of \(x\) is defined to be the largest power of \(p\) which divides the coefficients of \(x\), where \(x\) is considered as a polynomial in \({\mathbb{Z}}_p[x]/\langle D\rangle\). \(x\) is a unit if and only if the valuation of \(x\) is zero. Because of the valuation defined on them, Galois rings are Euclidean rings, so they may be used in Magma in any place where general Euclidean rings are valid. This includes many matrix and module functions, and the computation of Gröbner bases. Linear codes over Galois rings will be supported in the near future.