# Introduction

Magma supports finite extensions of the ring ${\mathbb{Z}}_p$ of $p$-adic integers or the field ${\mathbb{Q}}_p$ of $p$-adic numbers. Within this chapter, we mean these objects if we refer to local rings or fields. Section [Background](background.md#loc-background) provides more background information on the theory behind the $p$-adics.

Magma has three different models for working with these locals: fixed precision rings (`RngPadRes` and `RngPadResExt`, with element types `RngPadResElt` and `RngPadResExtElt`), free precision rings (`RngPad` and `FldPad`, with element types `RngPadElt` and `FldPadElt`) and exact $p$-adic rings (`RngXPad` and `FldXPad` with element types `RngXPadElt` and `FldXPadElt`). The merits of each model are discussed in Section [$p$-adic Rings](overview.md#loc-local-rings).

Magma also contains a type of local field where extensions can be made by any irreducible polynomial. For more information on these local fields, see Chapter [General $p$-Adic Extensions](../GeneralPAdicExtensions/index-general-padic-extensions.md#locala).