A local field is a type of field in mathematics that is complete with respect to a certain valuation and has a finite residue field. This means that it contains all the limits of sequences of elements that are "close" to each other according to the valuation. Local fields are important in number theory and algebraic geometry, as they provide a framework for studying solutions to equations over these fields. Examples of local fields include the field of p-adic numbers, denoted as ℚₚ, and the field of formal Laurent series over a finite field. Local fields allow mathematicians to analyze problems in a more manageable way by focusing on local properties rather than global ones.