Field (Mathematics)
A field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) are well-defined and satisfy the familiar properties of arithmetic. Formally, a field is a set equipped with two binary operations — addition and multiplication — such that:
- The set forms an abelian group under addition, with additive identity 0.
- The non-zero elements form an abelian group under multiplication, with multiplicative identity 1.
- Multiplication distributes over addition.
The rational numbers, real numbers, and complex numbers are all fields. Finite fields — fields with finitely many elements, also called Galois fields — are essential in coding theory and cryptography.
Fields are the natural setting for linear algebra, polynomial equations, and much of number theory. The study of fields and their extensions is the subject of Galois theory, which connects field structure to group symmetry.