Local Rings and Fields

Introduction

The most commonly encountered local fields are the field of p-adic numbers and the field k((t)) of power series over a finite field k. In addition, finite extensions of these fields may be constructed. In algebraic number theory, local fields arise as completions of “global” number fields and function fields. The most familiar example occurs in the case of the field of rational numbers; its completions consist of the field of real numbers and the field of p-adic numbers p for each prime p. Local fields are an indispensable tool in the theory of fields, much of which is developed by analysing their “local properties”. This approach is mirrored in many algorithms.

Implementation

Magma includes several implementations of local rings and fields (p, p, k((t)), and finite extensions of these). The lowest level implementation is for speed, while the highest level is for convenience, allowing calculations to be performed in the most user-friendly way, where each element has its own relative precision. Relative precisions are also handled smoothly when passing between extensions. Beyond field arithmetic, the most important algorithms provided are those for factorisation (S. Pauli) and roots of polynomials over local fields. One of the implementations of p-adic rings and fields is discussed in the next subsection.

Exact p-adic Rings and Fields

Exact p-adic rings and fields have been implemented by C. Doris. Such structures and their elements can be approximated to have as much precision as the objects they depend upon allow, sometimes to arbitrary precision. A “lazy” representation of elements is used whereby a record of their construction is kept but an approximation of an element is only computed when needed.

Elements of exact p-adic rings and fields have additional intrinsics which apply to them but not to elements of ordinary p-adic rings and fields. These intrinsics mitigate the infinite nature of the valuation intrinsic for exact zeros.

A polynomial ring type defined over an exact p-adic ring or field is available. Its elements are referred to as “exact polynomials”. Also an ordinary polynomial ring can be created over these exact p-adic rings and fields. Polynomials can be converted to “exact polynomials” where the properties of being weakly zero or weakly equal are recognized. The factors and roots of an exact polynomial can be computed.

Applications

An important application of local fields is to the calculation of a maximal order for a number field or a function field. Once the discriminant of the field has been factored, finding a maximal order can be reduced to “local calculations”. Local rings and fields have many applications outside algebraic number theory. Localisation is an important tool in algebra, especially in algebraic geometry. In practice, this means performing calculations in (some finite approximation of) a complete ring. For example, analysis of singularities on curves and surfaces typically involves calculating in power series rings. In a different direction, the most efficient methods for counting points on curves are p-adic in nature and are implemented in Magma using p-adic rings. For this application the speed of the p-adic arithmetic is critical.