Local fields

Local fields are mathematical systems used to study numbers and their properties in a localized setting. Think of them as specialized “number environments” focused on a particular prime number or a specific completion of a number system. They help mathematicians analyze how numbers behave locally—for example, examining solutions to equations in these fields can reveal insights about their behavior in broader contexts. Examples include the real numbers (focusing on infinity) or p-adic numbers (focusing on divisibility by a prime p). Local fields are fundamental in number theory, connecting global properties of numbers with local behavior.