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.

This article is about an algebraic structure. For vector valued functions, see Vector field. For other uses, see Field (disambiguation) § 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.

of addition and multiplication: $a + (b + c) = (a + b) + c$ , and $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ .

Associativity

of addition and multiplication: $a + b = b + a$ , and $a \cdot b = b \cdot a$ .

Commutativity

and multiplicative identity: there exist two distinct elements $0$ and $1$ in $F$ such that $a + 0 = a$ and $a \cdot 1 = a$ .

Additive

: 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 inverses

: for every $a \neq 0$ in $F$ , there exists an element in $F$ , denoted by $a -1$ or $1/ a$ , called the multiplicative inverse of $a$ , such that $a \cdot a -1 = 1$ .

Multiplicative inverses

of multiplication over addition: $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$ .

Distributivity

Constructing fields[edit]

Constructing fields from rings[edit]

A commutative ring is a set that is equipped with an addition and multiplication operation and satisfes all the axioms of a field, except for the existence of multiplicative inverses $a -1$ .^[26] For example, the integers $Z$ form a commutative ring, but not a field: the reciprocal of an integer $n$ is not itself an integer, unless $n = \pm1$ .

In the hierarchy of algebraic structures fields can be characterized as the commutative rings $R$ in which every nonzero element is a unit (which means every element is invertible). Similarly, fields are the commutative rings with precisely two distinct ideals, $(0)$ and $R$ . Fields are also precisely the commutative rings in which $(0)$ is the only prime ideal.

Given a commutative ring $R$ , there are two ways to construct a field related to $R$ , i.e., two ways of modifying $R$ such that all nonzero elements become invertible: forming the field of fractions, and forming residue fields. The field of fractions of $Z$ is $Q$ , the rationals, while the residue fields of $Z$ are the finite fields $F p$ .

finite extensions of $Q p$ (local fields of characteristic zero)

finite extensions of $F p ((t))$ , the field of Laurent series over $F p$ (local fields of characteristic $p$ ).

Applications[edit]

Linear algebra and commutative algebra[edit]

If $a \neq 0$ , then the equation