Field (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. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.

The best known fields are the field of rational numbers, the field of real numbers, and the field of complex numbers. 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. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements.

The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straightedge. 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.

Fields serve as foundational notions in several mathematical domains. This includes different branches of mathematical analysis, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. 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. Number fields, the siblings of the field of rational numbers, are studied in depth in number theory. Function fields can help describe properties of geometric objects.

Informally, a field is a set, along with two operations defined on that set: an addition operation a + b and a multiplication operation a ⋅ b, both of which behave similarly as they do for rational numbers and real numbers. This includes the existence of an additive inverse −a for each element a and of a multiplicative inverse b⁻¹ for each nonzero element b. This allows the definition of the so-called inverse operations, subtraction a − b and division a / b, as a − b = a + (−b) and a / b = a ⋅ b⁻¹. Often the product a ⋅ b is represented by juxtaposition, as ab.

Formally, a field is a set F together with two binary operations on F called addition and multiplication. 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. The result of the addition of a and b is called the sum of a and b, and is denoted a + b. Similarly, the result of the multiplication of a and b is called the product of a and b, and is denoted a ⋅ b. These operations are required to satisfy the following properties, referred to as field axioms.

These axioms are required to hold for all elements a, b, c of the field F:

An 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 form a group under multiplication with 1 as the multiplicative identity; and multiplication distributes over addition.