A Note on a Standard Model for Galois Rings

TL;DR

This paper introduces a standard model for Galois rings built upon the standard model of their residual fields, providing an algorithmic way to generate all Galois rings of a given size.

Contribution

It proposes a novel standard model for Galois rings based on residual fields, enabling systematic construction of Galois rings with specified characteristics.

Findings

Provides an explicit algorithm for constructing Galois rings.

Establishes a standard model linking Galois rings to their residual fields.

Facilitates systematic study of Galois rings through this model.

Abstract

In this work, we present a standard model for Galois rings based on the standard model of their residual fields, that is, a sequence of Galois rings starting with ${\mathbb Z}_{p^r} that coves all the Galois rings with that characteristic ring and such that there is an algorithm producing each member of the sequence whose input is the size of the required ring.