Magma supports finite extensions of the ring Zp of p-adic integers or the field Qp 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.