CS Knowledge Base

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 that 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 $a + b$ and a multiplication operation $a \cdot b$ that behave as they do for rational numbers and real numbers. The requirements include 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 $a / b = a \cdot b -1$ . Often the product $a \cdot b$ is represented by juxtaposition, as $ab$ .

Classic definition

Formally, a field is a set $F$ together with two binary operations on $F$ , called addition and multiplication, satisfying the axioms given below.^[1] A binary operation on $F$ is a mapping $F \times F \to F$ ; it sends each ordered pair of elements of $F$ to a uniquely determined element of $F$ .^[2]^[3] The result of the addition of $a$ and $b$ is called the sum of $a$ and $b$ , and is denoted $a + b$ . The result of the multiplication of $a$ and $b$ is called the product of $a$ and $b$ , and is denoted $a \cdot b$ . These operations are required to satisfy the following properties, called field axioms.

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

Associativity of addition and multiplication: $a + (b + c) = (a + b) + c$ , and $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ .
Commutativity of addition and multiplication: $a + b = b + a$ , and $a \cdot b = b \cdot a$ .
Additive and multiplicative identity: there exist distinct elements $0$ and $1$ in $F$ such that $a + 0 = a$ and $a \cdot 1 = a$ .
Additive inverses: for every $a$ in $F$ , there exists an element in $F$ , denoted $- a$ , called the additive inverse of $a$ , such that $a + (- a) = 0$ .
Multiplicative 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 $a \cdot a -1 = 1$ .
Distributivity of multiplication over addition: $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$ .

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 $0$ ;
the nonzero elements form a group under multiplication; and
multiplication distributes over addition.

Even more succinctly: a field is a commutative ring in which $0 \neq 1$ and all nonzero elements are invertible under multiplication.

Alternative definitions

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.^[4] 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 $1$ ). These operations are then subject to the conditions above. Avoiding existential quantifiers is important in constructive mathematics and computing.^[5] One may equivalently define a field by the same two binary operations, one unary operation (the multiplicative inverse), and two (not necessarily distinct) constants $1$ and $-1$ , since $0 = 1 + (-1)$ and $- a = (-1) a$ .^[a]

Examples

Rational numbers

Rational numbers were widely used for a long time before the development of fields. They are numbers that can be written as fractions $a / b$ , where $a$ and $b$ are integers, and $b \neq 0$ . The additive inverse of such a fraction is $- a / b$ , and the multiplicative inverse (provided that $a \neq 0$ ) is $b / a$ , which can be seen as follows:

{\frac {b}{a}}\cdot {\frac {a}{b}}={\frac {ba}{ab}}=1.

The abstractly required field axioms reduce to standard properties of rational numbers. For example, the law of distributivity can be proven as follows:^[6]

{\begin{aligned}&{\frac {a}{b}}\cdot \left({\frac {c}{d}}+{\frac {e}{f}}\right)\\[6pt]={}&{\frac {a}{b}}\cdot \left({\frac {c}{d}}\cdot {\frac {f}{f}}+{\frac {e}{f}}\cdot {\frac {d}{d}}\right)\\[6pt]={}&{\frac {a}{b}}\cdot \left({\frac {cf}{df}}+{\frac {ed}{fd}}\right)={\frac {a}{b}}\cdot {\frac {cf+ed}{df}}\\[6pt]={}&{\frac {a(cf+ed)}{bdf}}={\frac {acf}{bdf}}+{\frac {aed}{bdf}}={\frac {ac}{bd}}+{\frac {ae}{bf}}\\[6pt]={}&{\frac {a}{b}}\cdot {\frac {c}{d}}+{\frac {a}{b}}\cdot {\frac {e}{f}}.\end{aligned}}

Real and complex numbers

The real numbers $R$ , with the usual operations of addition and multiplication, also form a field. The complex numbers $C$ consist of expressions

a + bi,

with

a, b

real,

where $i$ is the imaginary unit, i.e., a (non-real) number satisfying $i 2 = -1$ . 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 $C$ . For example, the distributive law enforces

(a + bi)(c + di) = ac + bci + adi + bdi 2 = (ac - bd) + (bc + ad) i .

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.^[7] 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 $Q$ of rational numbers. The illustration shows the construction of square roots of constructible numbers, not necessarily contained within $Q$ . Using the labeling in the illustration, construct the segments $AD$ , $DB$ , and a semicircle over $AB$ (center at the midpoint $O$ ), which intersects the perpendicular line through $D$ in a point $C$ , at a distance of exactly $h={\sqrt {p}}$ from $B$ when $BD$ has length one.

Not all real numbers are constructible. It can be shown that ${\sqrt[{3}]{2}}$ 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

Addition

Multiplication

+	$O$	$I$	$A$	$B$
$O$	$O$	$I$	$A$	$B$
$I$	$I$	$O$	$B$	$A$
$A$	$A$	$B$	$O$	$I$
$B$	$B$	$A$	$I$	$O$

⋅	$O$	$I$	$A$	$B$
$O$	$O$	$O$	$O$	$O$
$I$	$O$	$I$	$A$	$B$
$A$	$O$	$A$	$B$	$I$
$B$	$O$	$B$	$I$	$A$

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 $O$ , $I$ , $A$ , and $B$ . The notation is chosen such that $O$ plays the role of the additive identity element (denoted 0 in the axioms above), and $I$ is the multiplicative identity (denoted $1$ in the axioms above). The field axioms can be verified by using some more field theory, or by direct computation. For example,

A \cdot (B + A) = A \cdot I = A

, which equals

A \cdot B + A \cdot A = I + B = A

, as required by the distributivity.

This field is called a finite field or Galois field with four elements, and is denoted $F 4$ or $GF(4)$ .^[8] The subset consisting of $O$ and $I$ (highlighted in red in the tables at the right) is also a field, known as the binary field $F 2$ or $GF(2)$ .

Elementary notions

In this section, $F$ denotes an arbitrary field and $a$ and $b$ are arbitrary elements of $F$ .

Consequences of the definition

One has $a \cdot 0 = 0$ and $- a = (-1) \cdot a$ .^[9]

If $ab = 0$ then $a$ or $b$ must be $0$ , since, if $a \neq 0$ , then $b = (a -1 a) b = a -1 (ab) = a -1 \cdot 0 = 0$ . This means that every field is an integral domain.

In addition, the following properties are true for any elements $a$ and $b$ :

-0 = 0

1 -1 = 1

-(- a) = a

(a -1) -1 = a

if

a \neq 0

(- a) \cdot b = a \cdot (- b) = -(a \cdot b)

Additive and multiplicative groups of a field

The axioms of a field $F$ imply that it is an abelian group under addition. This group is called the additive group of the field, and is sometimes denoted by $(F, +)$ when denoting it simply as $F$ could be confusing.

Similarly, the nonzero elements of $F$ form an abelian group under multiplication, called the multiplicative group, and denoted by $(F\smallsetminus \{0\},\cdot )$ or just $F\smallsetminus \{0\}$ , or $F \times$ .

A field may thus be defined as set $F$ equipped with two operations denoted as an addition and a multiplication such that $F$ is an abelian group under addition, $F\smallsetminus \{0\}$ is an abelian group under multiplication (where 0 is the identity element of the addition), and multiplication is distributive over addition.^[b] Some elementary statements about fields can therefore be obtained by applying general facts of groups. For example, the additive and multiplicative inverses $- a$ and $a -1$ are uniquely determined by $a$ .

The requirement $1 \neq 0$ is imposed by convention to exclude the trivial ring, which consists of a single element; indeed, the nonzero elements of the trivial ring (there are none) do not form a group, since a group must have at least one element.^[c]

Every finite subgroup of the multiplicative group of a field is cyclic (see Root of unity § Cyclic groups).

Characteristic

In addition to the multiplication of two elements of $F$ , it is possible to define the product $n \cdot a$ of an arbitrary element $a$ of $F$ by a positive integer $n$ to be the $n$ -fold sum

a + a + ... + a

(which is an element of

F

.)

If there is no positive integer such that

n \cdot 1 = 0

,

then $F$ is said to have characteristic $0$ .^[11] For example, the field of rational numbers $Q$ has characteristic 0 since no positive integer $n$ is zero. Otherwise, if there is a positive integer $n$ satisfying this equation, the smallest such positive integer can be shown to be a prime number. It is usually denoted by $p$ and the field is said to have characteristic $p$ then. For example, the field $F 4$ has characteristic $2$ since (in the notation of the above addition table) $I + I = O$ .

If $F$ has characteristic $p$ , then $p \cdot a = 0$ for all $a$ in $F$ . This implies that

(a + b) p = a p + b p

,

since all other binomial coefficients appearing in the binomial formula are divisible by $p$ . Here, $a p := a \cdot a \cdot \dots \cdot a$ ( $p$ factors) is the $p$ th power, i.e., the $p$ -fold product of the element $a$ . Therefore, the Frobenius map

F \to F : x \mapsto x p

is compatible with the addition in $F$ (and also with the multiplication), and is therefore a field homomorphism.^[12] The existence of this homomorphism makes fields in characteristic $p$ quite different from fields of characteristic $0$ .

Subfields and prime fields

A subfield $E$ of a field $F$ is a subset of $F$ that is a field with respect to the field operations of $F$ . Equivalently $E$ is a subset of $F$ that contains $1$ , and is closed under addition, multiplication, additive inverse and multiplicative inverse of a nonzero element. This means that $1 ∊ E$ , that for all $a, b ∊ E$ both $a + b$ and $a \cdot b$ are in $E$ , and that for all $a \neq 0$ in $E$ , both $- a$ and $1/ a$ are in $E$ .

Field homomorphisms are maps $φ : E \to F$ between two fields such that $φ (e 1 + e 2) = φ (e 1) + φ (e 2)$ , $φ (e 1 e 2) = φ (e 1) φ (e 2)$ , and $φ (1 E) = 1 F$ , where $e 1$ and $e 2$ are arbitrary elements of $E$ . All field homomorphisms are injective.^[13] If $φ$ is also surjective, it is called an isomorphism (or the fields $E$ and $F$ are called isomorphic).

A field is called a prime field if it has no proper (i.e., strictly smaller) subfields. Any field $F$ contains a prime field. If the characteristic of $F$ is $p$ (a prime number), the prime field is isomorphic to the finite field $F p$ introduced below. Otherwise the prime field is isomorphic to $Q$ .^[14]

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 $F 4$ is a field with four elements. Its subfield $F 2$ is the smallest field, because by definition a field has at least two distinct elements, $0$ and $1$ .

The simplest finite fields, with prime order, are most directly accessible using modular arithmetic. For a fixed positive integer $n$ , arithmetic "modulo $n$ " means to work with the numbers

Z / n Z = {0, 1, ..., n - 1}.

The addition and multiplication on this set are done by performing the operation in question in the set $Z$ of integers, dividing by $n$ and taking the remainder as result. This construction yields a field precisely if $n$ is a prime number. For example, taking the prime $n = 2$ results in the above-mentioned field $F 2$ . For $n = 4$ and more generally, for any composite number (i.e., any number $n$ which can be expressed as a product $n = r \cdot s$ of two strictly smaller natural numbers), $Z / n Z$ is not a field: the product of two non-zero elements is zero since $r \cdot s = 0$ in $Z / n Z$ , which, as was explained above, prevents $Z / n Z$ from being a field. The field $Z / p Z$ with $p$ elements ( $p$ being prime) constructed in this way is usually denoted by $F p$ .

Every finite field $F$ has $q = p n$ elements, where $p$ is prime and $n \geq 1$ . This statement holds since $F$ may be viewed as a vector space over its prime field. The dimension of this vector space is necessarily finite, say $n$ , which implies the asserted statement.^[15]

A field with $q = p n$ elements can be constructed as the splitting field of the polynomial

f (x) = x q - x

.

Such a splitting field is an extension of $F p$ in which the polynomial $f$ has $q$ zeros. This means $f$ has as many zeros as possible since the degree of $f$ is $q$ . For $q = 2 2 = 4$ , it can be checked case by case using the above multiplication table that all four elements of $F 4$ satisfy the equation $x 4 = x$ , so they are zeros of $f$ . By contrast, in $F 2$ , $f$ has only two zeros (namely $0$ and $1$ ), so $f$ 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.^[16] It is thus customary to speak of the finite field with $q$ elements, denoted by $F q$ or $GF(q)$ .

History

Historically, three algebraic disciplines led to the concept of a field: the question of solving polynomial equations, algebraic number theory, and algebraic geometry.^[17] A first step towards the notion of a field was made in 1770 by Joseph-Louis Lagrange, who observed that permuting the zeros $x 1, x 2, x 3$ of a cubic polynomial in the expression

(x 1 + ωx 2 + ω 2 x 3) 3

(with $ω$ being a third root of unity) only yields two values. This way, Lagrange conceptually explained the classical solution method of Scipione del Ferro and François Viète, which proceeds by reducing a cubic equation for an unknown $x$ to a quadratic equation for $x 3$ .^[18] Together with a similar observation for equations of degree 4, Lagrange thus linked what eventually became the concept of fields and the concept of groups.^[19] Vandermonde, also in 1770, and to a fuller extent, Carl Friedrich Gauss, in his Disquisitiones Arithmeticae (1801), studied the equation

x p = 1

for a prime $p$ and, again using modern language, the resulting cyclic Galois group. Gauss deduced that a regular $p$ -gon can be constructed if $p = 2 2 k + 1$ . Building on Lagrange's work, Paolo Ruffini claimed (1799) that quintic equations (polynomial equations of degree $5$ ) cannot be solved algebraically; however, his arguments were incomplete. These gaps were filled by Niels Henrik Abel in 1824.^[20] Évariste Galois, in 1832, devised necessary and sufficient criteria for a polynomial equation to be algebraically solvable, thus establishing in effect what is known as Galois theory today. Both Abel and Galois worked with what is today called an algebraic number field, but they conceived neither an explicit notion of a field, nor of a group.

In 1871 Richard Dedekind introduced, for a set of real or complex numbers that is closed under the four arithmetic operations, the German word Körper, which means "body" or "corpus" (to suggest an organically closed entity). The English term "field" was introduced by Moore (1893).^[21]

[1]

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[3]

[4]

[5]

[a]

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[7]

[8]

[9]

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[c]

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[21]

Field	Metric	Completion	zero sequence
$Q$	$\| x - y \|$ (usual absolute value)	R	$1/ n$
$Q$	obtained using the p-adic valuation, for a prime number $p$	$Q p$ ( $p$ -adic numbers)	$p n$
$F (t)$ ( $F$ any field)	obtained using the $t$ -adic valuation	$F ((t))$	$t n$

CS Knowledge Base

Definition

Classic definition

Alternative definitions

Examples

Rational numbers

Real and complex numbers

Constructible numbers

A field with four elements

Elementary notions

Consequences of the definition

Additive and multiplicative groups of a field

Characteristic

Subfields and prime fields

Finite fields

History

Constructing fields

Constructing fields from rings

Field of fractions

Residue fields

Constructing fields within a bigger field

Field extensions

Algebraic extensions

Transcendence bases

Closure operations

Fields with additional structure

Ordered fields

Topological fields

Local fields

Differential fields

Galois theory

Invariants of fields

Model theory of fields

Absolute Galois group

K-theory

Applications

Linear algebra and commutative algebra

Finite fields: cryptography and coding theory

Geometry: field of functions

Number theory: global fields

Related notions

Division rings

Notes

Citations

References

External links