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 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.

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.