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.

Definition

Informally, a field is a set, along with two operations defined on that set: an addition operation and a multiplication operation , both of which behave similarly as they do for rational numbers and real numbers. This includes the existence of an additive inverse for all elements and of a multiplicative inverse for every 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''. A binary operation on is a mapping , that is, a correspondence that associates with each ordered pair of elements of a uniquely determined element of . The result of the addition of and is called the sum of and , and is denoted . Similarly, 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, referred to as '' 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 two 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, and more succinct, definition is: a field has two commutative operations, called addition and multiplication; it is a group under addition with as the additive identity; the nonzero elements form a group under multiplication with as the multiplicative identity; and multiplication distributes over addition.

Even more succinctly: a field is a commutative ring where 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 (addition, subtraction, multiplication, and division) 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 (addition and multiplication), two unary operations (yielding the additive and multiplicative inverses respectively), and two nullary operations (the constants and ). 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 (the multiplicative inverse), and two (not necessarily distinct) 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 (provided that ) is , which can be seen as follows:
: $\frac b a \cdot \frac a b = \frac = 1.$

The abstractly required field axioms reduce to standard properties of rational numbers. For example, the law of distributivity can be proven as follows:
:
\begin
& \frac a b \cdot \left(\frac c d + \frac e f \right) \\ pt= & \frac a b \cdot \left(\frac c d \cdot \frac f f + \frac e f \cdot \frac d d \right) \\ pt= & \frac \cdot \left(\frac + \frac\right) = \frac \cdot \frac \\ pt= & \frac = \frac + \frac = \frac + \frac \\ pt= & \frac a b \cdot \frac c d + \frac a b \cdot \frac e f.
\end

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
: with real,
where is the imaginary unit, i.e., a (non-real) 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 (adding the Cartesian coordinates), and the multiplication is – less intuitively – combining rotating and scaling of the arrows (adding the angles and multiplying the lengths). 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 (in)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 (center at the midpoint ), which intersects the perpendicular line through in a point , at a distance of exactly $h=\sqrt p$ from when has length one.

Not all real numbers are constructible. It can be shown that \sqrt 2 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

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 (denoted 0 in the axioms above), and is the multiplicative identity (denoted in the axioms above). The field axioms can be verified by using some more field theory, or by direct computation. For example,
: , which equals , as required by the distributivity.

This field is called a finite field or Galois field with four elements, and is denoted or . The subset consisting of and (highlighted in red in the tables at the right) 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 . In particular, one may deduce the additive inverse of every element as soon as one knows .

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 :
:
:
:
:
: if

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 $(F \smallsetminus \, \cdot)$ or just $F \smallsetminus \$, 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, $F \smallsetminus \$ is an abelian group under multiplication (where 0 is the identity element of the addition), 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; this guides any choice of the axioms that define fields.

Every finite subgroup of the multiplicative group of a field is cyclic (see ').

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
: (which is an element of .)

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 (in the notation of the above addition table) .

If has characteristic , then for all in . This implies that
: ,
since all other binomial coefficients appearing in the binomial formula are divisible by . Here, ( factors) is the th power, i.e., the -fold product of the element . Therefore, the Frobenius map
:
is compatible with the addition in (and also with the multiplication), and is therefore a field homomorphism. The existence of this homomorphism makes fields in characteristic quite different from fields of characteristic .

Subfields and prime fields

A '' subfield'' of a field is a subset of that is a field with respect to the field operations of . Equivalently is a subset of that contains , and is closed under addition, multiplication, additive inverse and multiplicative inverse of a nonzero element. This means that , that for all both and are in , and that for all in , both and are in .

Field homomorphisms are maps between two fields such that , , and , where and are arbitrary elements of . All field homomorphisms are injective. If is also surjective, it is called an isomorphism (or the fields and are called isomorphic).

A field is called a prime field if it has no proper (i.e., strictly smaller) subfields. Any field contains a prime field. If the characteristic of is (a prime number), the prime field is isomorphic to the finite field introduced below. Otherwise the prime field is isomorphic to .

Finite fields

''Finite fields'' (also called ''Galois fields'') are fields with finitely many elements, whose number is also referred to as the order of the field. The above introductory example is a field with four elements. Its subfield is the smallest field, because by definition a field has at least two distinct elements, and .

The simplest finite fields, with prime order, are most directly accessible using modular arithmetic. For a fixed positive integer , arithmetic "modulo " means to work with the numbers
:
The addition and multiplication on this set are done by performing the operation in question in the set of integers, dividing by and taking the remainder as result. This construction yields a field precisely if is a prime number. For example, taking the prime results in the above-mentioned field . For and more generally, for any composite number (i.e., any number which can be expressed as a product of two strictly smaller natural numbers), is not a field: the product of two non-zero elements is zero since in , which, as was explained above, prevents from being a field. The field with elements ( being prime) constructed in this way is usually denoted by .

Every finite field has elements, where is prime and . This statement holds since may be viewed as a vector space over its prime field. The dimension of this vector space is necessarily finite, say , which implies the asserted statement.

A field with elements can be constructed as the splitting field of the polynomial
: .
Such a splitting field is an extension of in which the polynomial has zeros. This means has as many zeros as possible since the degree of is . For , it can be checked case by case using the above multiplication table that all four elements of satisfy the equation , so they are zeros of . By contrast, in , has only two zeros (namely and ), so does not split into linear factors in this smaller field. Elaborating further on basic field-theoretic notions, it can be shown that two finite fields with the same order are isomorphic. It is thus customary to speak of ''the'' finite field with elements, denoted by or .

History

Historically, three algebraic disciplines led to the concept of a field: the question of solving polynomial equations, algebraic number theory, and algebraic geometry. A first step towards the notion of a field was made in 1770 by Joseph-Louis Lagrange

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