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A '''field''' is an [[Abstract Algebra|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|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|coding theory]] and [[Cryptography|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|Galois theory]], which connects field structure to group symmetry.

== See also ==

* [[Abstract Algebra]]
* [[Ring Theory]]
* [[Galois Theory]]
* [[Field Theory]]
* [[Finite Field]]
* [[Vector Space]]