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 numbers do. 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, finite fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry.
The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straightedge alone. 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. Finite fields are used for error correction codes and cryptography.

Definition

Informally, a field is a set with an addition operation and a multiplication operation that behave as they do for rational numbers and real numbers. The requirements include the existence of an additive inverse for each element and of a multiplicative inverse for each nonzero element. This allows the definition of the so-called inverse operations, subtraction and division, as and.
Often the product is represented by juxtaposition, as.

Classic definition

Formally, a field is a set together with two binary operations on, called addition and multiplication, satisfying the axioms given below. A binary operation on is a mapping ; it sends each ordered pair of elements of to a uniquely determined element of. The result of the addition of and is called the sum of and, and is denoted. The result of the multiplication of and is called the product of and, and is denoted. These operations are required to satisfy the following properties, called field axioms.
These axioms are required to hold for all elements ,, of the field :

Associativity of addition and multiplication:, and.
Commutativity of addition and multiplication:, and.
Additive and multiplicative identity: there exist distinct elements and in such that and.
Additive inverses: for every in, there exists an element in, denoted, called the additive inverse of, such that.
Multiplicative inverses: for every in, there exists an element in, denoted by or, called the multiplicative inverse of, such that.
Distributivity of multiplication over addition:.

An equivalent but more succinct definition is: a field is a set with two commutative operations, called addition and multiplication, such that

it is a group under addition, with additive identity called ;
the nonzero elements form a group under multiplication; and
multiplication distributes over addition.

Even more succinctly: a field is a commutative ring in which and all nonzero elements are invertible under multiplication.

Alternative definition

Fields can also be defined in different, but equivalent, ways. One can alternatively define a field by four binary operations and their required properties. Division by zero is, by definition, excluded. In order to avoid existential quantifiers, fields can be defined by two binary operations, two unary operations, and two nullary operations. These operations are then subject to the conditions above. Avoiding existential quantifiers is important in constructive mathematics and computing. One may equivalently define a field by the same two binary operations, one unary operation, and two constants and, since and.

Examples

Rational numbers

Rational numbers have been widely used a long time before the elaboration of the concept of field.
They are numbers that can be written as fractions
, where and are integers, and. The additive inverse of such a fraction is, and the multiplicative inverse is, which can be seen as follows:
The abstractly required field axioms reduce to standard properties of rational numbers. For example, the law of distributivity can be proven as follows:

Real and complex numbers

The real numbers, with the usual operations of addition and multiplication, also form a field. The complex numbers consist of expressions
where is the imaginary unit, i.e., a number satisfying.
Addition and multiplication of real numbers are defined in such a way that expressions of this type satisfy all field axioms and thus hold for. For example, the distributive law enforces
It is immediate that this is again an expression of the above type, and so the complex numbers form a field. Complex numbers can be geometrically represented as points in the plane, with Cartesian coordinates given by the real numbers of their describing expression, or as the arrows from the origin to these points, specified by their length and an angle enclosed with some distinct direction. Addition then corresponds to combining the arrows to the intuitive parallelogram, and the multiplication is – less intuitively – combining rotating and scaling of the arrows. The fields of real and complex numbers are used throughout mathematics, physics, engineering, statistics, and many other scientific disciplines.

Constructible numbers

In antiquity, several geometric problems concerned the feasibility of constructing certain numbers with compass and straightedge. For example, it was unknown to the Greeks that it is, in general, impossible to trisect a given angle in this way. These problems can be settled using the field of constructible numbers. Real constructible numbers are, by definition, lengths of line segments that can be constructed from the points 0 and 1 in finitely many steps using only compass and straightedge. These numbers, endowed with the field operations of real numbers, restricted to the constructible numbers, form a field, which properly includes the field of rational numbers. The illustration shows the construction of square roots of constructible numbers, not necessarily contained within. Using the labeling in the illustration, construct the segments,, and a semicircle over , which intersects the perpendicular line through in a point, at a distance of exactly from when has length one.
Not all real numbers are constructible. It can be shown that is not a constructible number, which implies that it is impossible to construct with compass and straightedge the length of the side of a cube with volume 2, another problem posed by the ancient Greeks.

A field with four elements

Addition

Multiplication

In addition to familiar number systems such as the rationals, there are other, less immediate examples of fields. The following example is a field consisting of four elements called,,, and. The notation is chosen such that plays the role of the additive identity element, and is the multiplicative identity. The field axioms can be verified by using some more field theory, or by direct computation. For example,
This field is called a finite field or Galois field with four elements, and is denoted or. The subset consisting of and is also a field, known as the binary field or.

Elementary notions

In this section, denotes an arbitrary field and and are arbitrary elements of.

Consequences of the definition

One has and.
If then or must be, since, if, then
. This means that every field is an integral domain.
In addition, the following properties are true for any elements and :

Additive and multiplicative groups of a field

The axioms of a field imply that it is an abelian group under addition. This group is called the additive group of the field, and is sometimes denoted by when denoting it simply as could be confusing.
Similarly, the nonzero elements of form an abelian group under multiplication, called the multiplicative group, and denoted by or just, or.
A field may thus be defined as set equipped with two operations denoted as an addition and a multiplication such that is an abelian group under addition, is an abelian group under multiplication, and multiplication is distributive over addition. Some elementary statements about fields can therefore be obtained by applying general facts of groups. For example, the additive and multiplicative inverses and are uniquely determined by.
The requirement is imposed by convention to exclude the trivial ring, which consists of a single element; indeed, the nonzero elements of the trivial ring do not form a group, since a group must have at least one element.
Every finite subgroup of the multiplicative group of a field is cyclic.

Characteristic

In addition to the multiplication of two elements of, it is possible to define the product of an arbitrary element of by a positive integer to be the -fold sum
If there is no positive integer such that
then is said to have characteristic. For example, the field of rational numbers has characteristic 0 since no positive integer is zero. Otherwise, if there is a positive integer satisfying this equation, the smallest such positive integer can be shown to be a prime number. It is usually denoted by and the field is said to have characteristic then.
For example, the field has characteristic since .
If has characteristic, then for all in. This implies that
since all other binomial coefficients appearing in the binomial formula are divisible by. Here, is the th power, i.e., the -fold product of the element. Therefore, the Frobenius map
is compatible with the addition in , and is therefore a field homomorphism. The existence of this homomorphism makes fields in characteristic quite different from fields of characteristic.