Avoiding existential quantifiers is important in constructive mathematics and computing. These operations are then subject to the conditions above. In order to avoid existential quantifiers, fields can be defined by two binary operations (addition and multiplication), two unary operations (yielding the additive and multiplicative inverses respectively), and two nullary operations (the constants 0 and 1). Division by zero is, by definition, excluded. One can alternatively define a field by four binary operations (addition, subtraction, multiplication, and division) and their required properties.

Distributivity of multiplication over addition: a ⋅ ( b + c) = ( a ⋅ b) + ( a ⋅ c).Īn equivalent, and more succinct, definition is: a field has two commutative operations, called addition and multiplication it is a group under addition with 0 as the additive identity the nonzero elements are a group under multiplication with 1 as the multiplicative identity and multiplication distributes over addition.Įven more succinctly: a field is a commutative ring where 0 ≠ 1 and all nonzero elements are invertible under multiplication.įields can also be defined in different, but equivalent ways.

Multiplicative inverses: for every a ≠ 0 in F, there exists an element in F, denoted by a −1 or 1/ a, called the multiplicative inverse of a, such that a ⋅ a −1 = 1.

Additive inverses: for every a in F, there exists an element in F, denoted − a, called the additive inverse of a, such that a + (− a) = 0.

Additive and multiplicative identity: there exist two distinct elements 0 and 1 in F such that a + 0 = a and a ⋅ 1 = a.

Commutativity of addition and multiplication: a + b = b + a, and a ⋅ b = b ⋅ a.

Associativity of addition and multiplication: a + ( b + c) = ( a + b) + c, and a ⋅ ( b ⋅ c) = ( a ⋅ b) ⋅ c.

These operations are required to satisfy the following properties, referred to as field axioms (in these axioms, a, b, and c are arbitrary elements of the field F): Similarly, the result of the multiplication of a and b is called the product of a and b, and is denoted ab or a ⋅ b. The result of the addition of a and b is called the sum of a and b, and is denoted a + b. A binary operation on F is a mapping F × F → F, that is, a correspondence that associates with each ordered pair of elements of F a uniquely determined element of F.

Classic definition įormally, a field is a set F together with two binary operations on F called addition and multiplication. This allows one to also consider the so-called inverse operations of subtraction, a − b, and division, a / b, by defining:Ī − b := a + (− b), a / b := a ⋅ b −1. Informally, a field is a set, along with two operations defined on that set: an addition operation written as a + b, and a multiplication operation written as a ⋅ b, both of which behave similarly as they behave for rational numbers and real numbers, including the existence of an additive inverse − a for all elements a, and of a multiplicative inverse b −1 for every nonzero element b. Function fields can help describe properties of geometric objects. Number fields, the siblings of the field of rational numbers, are studied in depth in number theory. Most importantly for algebraic purposes, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra.

Basic theorems in analysis hinge on the structural properties of the field of real numbers. This includes different branches of mathematical analysis, which are based on fields with additional structure. Galois theory, devoted to understanding the symmetries of field extensions, provides an elegant proof of the Abel-Ruffini theorem that general quintic equations cannot be solved in radicals.įields serve as foundational notions in several mathematical domains. The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straightedge.

Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers.

A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do.

Not real numbers