Field (mathematics)

Diagram of symbols of arithmetic operations

Fields are an algebraic structure which are closed under the four usual arithmetic operations.

Template:Algebraic structures 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 Template:Math and a multiplication operation Template:Math, both of which behave similarly as they do for rational numbers and real numbers. This includes the existence of an additive inverse Template:Math for each element Template:Mvar and of a multiplicative inverse Template:Math for each nonzero element Template:Mvar. This allows the definition of the so-called inverse operations, subtraction Template:Math and division Template:Math, as Template:Math and Template:Math. Often the product Template:Math is represented by juxtaposition, as Template:Mvar.

Classic definition

Formally, a field is a set Template:Mvar together with two binary operations on Template:Mvar called addition and multiplication. A binary operation on Template:Mvar is a mapping Template:Math, that is, a correspondence that associates with each ordered pair of elements of Template:Mvar a uniquely determined element of Template:Mvar. The result of the addition of Template:Mvar and Template:Mvar is called the sum of Template:Mvar and Template:Mvar, and is denoted Template:Math. Similarly, the result of the multiplication of Template:Mvar and Template:Mvar is called the product of Template:Mvar and Template:Mvar, and is denoted Template:Math. These operations are required to satisfy the following properties, referred to as field axioms.

These axioms are required to hold for all elements Template:Mvar, Template:Mvar, Template:Mvar of the field Template:Mvar:

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

An equivalent, and more succinct, definition is: a field has two commutative operations, called addition and multiplication; it is a group under addition with Template:Math as the additive identity; the nonzero elements form a group under multiplication with Template:Math as the multiplicative identity; and multiplication distributes over addition.

Even more succinctly: a field is a commutative ring where Template:Math 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 Template:Math and Template:Math). 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 Template:Math and Template:Math, since Template:Math and Template:Math.Template:Efn

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 Template:Math, where Template:Mvar and Template:Mvar are integers, and Template:Math. The additive inverse of such a fraction is Template:Math, and the multiplicative inverse (provided that Template:Math) is Template:Math, 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:

$\begin{array}{rl}& \frac{a}{b}\cdot (\frac{c}{d}+\frac{e}{f})\\ =& \frac{a}{b}\cdot (\frac{c}{d}\cdot \frac{f}{f}+\frac{e}{f}\cdot \frac{d}{d})\\ =& \frac{a}{b}\cdot (\frac{cf}{df}+\frac{ed}{fd})=\frac{a}{b}\cdot \frac{cf+ed}{df}\\ =& \frac{a(cf+ed)}{bdf}=\frac{acf}{bdf}+\frac{aed}{bdf}=\frac{ac}{bd}+\frac{ae}{bf}\\ =& \frac{a}{b}\cdot \frac{c}{d}+\frac{a}{b}\cdot \frac{e}{f}.\end{array}$

Real and complex numbers

File:Complex multi.svg

The multiplication of complex numbers can be visualized geometrically by rotations and scalings.

The real numbers Template:Math, with the usual operations of addition and multiplication, also form a field. The complex numbers Template:Math consist of expressions

Template:Math with Template:Math real,

where Template:Mvar is the imaginary unit, i.e., a (non-real) number satisfying Template:Math. 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 Template:Math. For example, the distributive law enforces

Template:Math

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

File:Root construction geometric mean5.svg

The geometric mean theorem asserts that Template:Math. Choosing Template:Math allows construction of the square root of a given constructible number Template:Mvar.

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 Template:Math of rational numbers. The illustration shows the construction of square roots of constructible numbers, not necessarily contained within Template:Math. Using the labeling in the illustration, construct the segments Template:Math, Template:Math, and a semicircle over Template:Math (center at the midpoint Template:Mvar), which intersects the perpendicular line through Template:Mvar in a point Template:Mvar, at a distance of exactly $h=\sqrt{p}$ from Template:Mvar when Template:Math 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
+ Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar	⋅ Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar Template:Mvar

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 Template:Mvar, Template:Mvar, Template:Mvar, and Template:Mvar. The notation is chosen such that Template:Mvar plays the role of the additive identity element (denoted 0 in the axioms above), and Template:Mvar is the multiplicative identity (denoted Template:Math in the axioms above). The field axioms can be verified by using some more field theory, or by direct computation. For example,

Template:Math, which equals Template:Math, as required by the distributivity.

This field is called a finite field or Galois field with four elements, and is denoted Template:Math or Template:Math. The subset consisting of Template:Mvar and Template:Mvar (highlighted in red in the tables at the right) is also a field, known as the binary field Template:Math or Template:Math.

Elementary notions

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

Consequences of the definition

One has Template:Math and Template:Math. In particular, one may deduce the additive inverse of every element as soon as one knows Template:Math.

If Template:Math then Template:Math or Template:Mvar must be Template:Math, since, if Template:Math, then Template:Math. This means that every field is an integral domain.

In addition, the following properties are true for any elements Template:Mvar and Template:Mvar:

Template:Math

Template:Math

Template:Math

Template:Math if Template:Math

Template:Math

Additive and multiplicative groups of a field

The axioms of a field Template:Mvar imply that it is an abelian group under addition. This group is called the additive group of the field, and is sometimes denoted by Template:Math when denoting it simply as Template:Mvar could be confusing.

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

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

The requirement Template:Math 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.Template:Efn

Every finite subgroup of the multiplicative group of a field is cyclic (see Template:Slink).

Characteristic

In addition to the multiplication of two elements of Template:Mvar, it is possible to define the product Template:Math of an arbitrary element Template:Mvar of Template:Mvar by a positive integer Template:Mvar to be the Template:Mvar-fold sum

Template:Math (which is an element of Template:Mvar.)

If there is no positive integer such that

Template:Math,

then Template:Mvar is said to have characteristic Template:Math. For example, the field of rational numbers Template:Math has characteristic 0 since no positive integer Template:Mvar is zero. Otherwise, if there is a positive integer Template:Mvar satisfying this equation, the smallest such positive integer can be shown to be a prime number. It is usually denoted by Template:Mvar and the field is said to have characteristic Template:Mvar then. For example, the field Template:Math has characteristic Template:Math since (in the notation of the above addition table) Template:Math.

If Template:Mvar has characteristic Template:Mvar, then Template:Math for all Template:Mvar in Template:Mvar. This implies that

Template:Math,

since all other binomial coefficients appearing in the binomial formula are divisible by Template:Mvar. Here, Template:Math (Template:Mvar factors) is the Template:Mvarth power, i.e., the Template:Mvar-fold product of the element Template:Mvar. Therefore, the Frobenius map

Template:Math

is compatible with the addition in Template:Mvar (and also with the multiplication), and is therefore a field homomorphism. The existence of this homomorphism makes fields in characteristic Template:Mvar quite different from fields of characteristic Template:Math.

Subfields and prime fields

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

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

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

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 Template:Math is a field with four elements. Its subfield Template:Math is the smallest field, because by definition a field has at least two distinct elements, Template:Math and Template:Math.

File:Clock group.svg

In modular arithmetic modulo 12, Template:Math since Template:Math in Template:Math, which divided by Template:Math leaves remainder Template:Math. However, Template:Math is not a field because Template:Math is not a prime number.

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

Template:Math

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

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

A field with Template:Math elements can be constructed as the splitting field of the polynomial

Template:Math.

Such a splitting field is an extension of Template:Math in which the polynomial Template:Mvar has Template:Mvar zeros. This means Template:Mvar has as many zeros as possible since the degree of Template:Mvar is Template:Mvar. For Template:Math, it can be checked case by case using the above multiplication table that all four elements of Template:Math satisfy the equation Template:Math, so they are zeros of Template:Mvar. By contrast, in Template:Math, Template:Mvar has only two zeros (namely Template:Math and Template:Math), so Template:Mvar 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 Template:Mvar elements, denoted by Template:Math or Template:Math.

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, who observed that permuting the zeros Template:Math of a cubic polynomial in the expression

Template:Math

(with Template:Math 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 Template:Mvar to a quadratic equation for Template:Math. 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. Vandermonde, also in 1770, and to a fuller extent, Carl Friedrich Gauss, in his Disquisitiones Arithmeticae (1801), studied the equation

Template:Math

for a prime Template:Mvar and, again using modern language, the resulting cyclic Galois group. Gauss deduced that a [[regular polygon|regular Template:Mvar-gon]] can be constructed if Template:Math. Building on Lagrange's work, Paolo Ruffini claimed (1799) that quintic equations (polynomial equations of degree Template:Math) cannot be solved algebraically; however, his arguments were flawed. These gaps were filled by Niels Henrik Abel in 1824. 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 Template:Harvp.

By a field we will mean every infinite system of real or complex numbers so closed in itself and perfect that addition, subtraction, multiplication, and division of any two of these numbers again yields a number of the system.

— Template:Comma separated entries

In 1881 Leopold Kronecker defined what he called a domain of rationality, which is a field of rational fractions in modern terms. Kronecker's notion did not cover the field of all algebraic numbers (which is a field in Dedekind's sense), but on the other hand was more abstract than Dedekind's in that it made no specific assumption on the nature of the elements of a field. Kronecker interpreted a field such as Template:Math abstractly as the rational function field Template:Math. Prior to this, examples of transcendental numbers were known since Joseph Liouville's work in 1844, until Charles Hermite (1873) and Ferdinand von Lindemann (1882) proved the transcendence of Template:Mvar and Template:Math, respectively.

The first clear definition of an abstract field is due to Template:Harvp. In particular, Heinrich Martin Weber's notion included the field Template:Math. Giuseppe Veronese (1891) studied the field of formal power series, which led Template:Harvp to introduce the field of Template:Mvar-adic numbers. Template:Harvp synthesized the knowledge of abstract field theory accumulated so far. He axiomatically studied the properties of fields and defined many important field-theoretic concepts. The majority of the theorems mentioned in the sections Galois theory, Constructing fields and Elementary notions can be found in Steinitz's work. Template:Harvp linked the notion of orderings in a field, and thus the area of analysis, to purely algebraic properties. Emil Artin redeveloped Galois theory from 1928 through 1942, eliminating the dependency on the primitive element theorem.

Constructing fields

Constructing fields from rings

A commutative ring is a set that is equipped with an addition and multiplication operation and satisfies all the axioms of a field, except for the existence of multiplicative inverses Template:Math. For example, the integers Template:Math form a commutative ring, but not a field: the reciprocal of an integer Template:Mvar is not itself an integer, unless Template:Math.

In the hierarchy of algebraic structures fields can be characterized as the commutative rings Template:Mvar 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, Template:Math and Template:Mvar. Fields are also precisely the commutative rings in which Template:Math is the only prime ideal.

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

Field of fractions

Given an integral domain Template:Mvar, its field of fractions Template:Math is built with the fractions of two elements of Template:Mvar exactly as Q is constructed from the integers. More precisely, the elements of Template:Math are the fractions Template:Math where Template:Mvar and Template:Mvar are in Template:Mvar, and Template:Math. Two fractions Template:Math and Template:Math are equal if and only if Template:Math. The operation on the fractions work exactly as for rational numbers. For example,

$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}.$

It is straightforward to show that, if the ring is an integral domain, the set of the fractions form a field.

The field Template:Math of the rational fractions over a field (or an integral domain) Template:Mvar is the field of fractions of the polynomial ring Template:Math. The field Template:Math of formal Laurent series

$\sum _{i=k}^{\mathrm{\infty }}{a}_i{x}^i(k\in \mathbb{\mathbb{Z}},a_i\in F)$

over a field Template:Mvar is the field of fractions of the ring Template:Math of formal power series (in which Template:Math). Since any Laurent series is a fraction of a power series divided by a power of Template:Mvar (as opposed to an arbitrary power series), the representation of fractions is less important in this situation, though.

Residue fields

In addition to the field of fractions, which embeds Template:Mvar injectively into a field, a field can be obtained from a commutative ring Template:Mvar by means of a surjective map onto a field Template:Mvar. Any field obtained in this way is a quotient Template:Math, where Template:Mvar is a maximal ideal of Template:Mvar. If Template:Mvar has only one maximal ideal Template:Mvar, this field is called the residue field of Template:Mvar.

The ideal generated by a single polynomial Template:Mvar in the polynomial ring Template:Math (over a field Template:Mvar) is maximal if and only if Template:Mvar is irreducible in Template:Mvar, i.e., if Template:Mvar cannot be expressed as the product of two polynomials in Template:Math of smaller degree. This yields a field

Template:Math

This field Template:Mvar contains an element Template:Mvar (namely the residue class of Template:Mvar) which satisfies the equation

Template:Math.

For example, Template:Math is obtained from Template:Math by adjoining the imaginary unit symbol Template:Mvar, which satisfies Template:Math, where Template:Math. Moreover, Template:Mvar is irreducible over Template:Math, which implies that the map that sends a polynomial Template:Math to Template:Math yields an isomorphism

$\mathbf{𝐑}[X]/(X^2+1)\stackrel{\cong }{⟶}\mathbf{𝐂}.$

Constructing fields within a bigger field

Fields can be constructed inside a given bigger container field. Suppose given a field Template:Mvar, and a field Template:Mvar containing Template:Mvar as a subfield. For any element Template:Mvar of Template:Mvar, there is a smallest subfield of Template:Mvar containing Template:Mvar and Template:Mvar, called the subfield of F generated by Template:Mvar and denoted Template:Math. The passage from Template:Mvar to Template:Math is referred to by adjoining an element to Template:Mvar. More generally, for a subset Template:Math, there is a minimal subfield of Template:Mvar containing Template:Mvar and Template:Mvar, denoted by Template:Math.

The compositum of two subfields Template:Mvar and Template:Math of some field Template:Mvar is the smallest subfield of Template:Mvar containing both Template:Mvar and Template:Math. The compositum can be used to construct the biggest subfield of Template:Mvar satisfying a certain property, for example the biggest subfield of Template:Mvar, which is, in the language introduced below, algebraic over Template:Mvar.Template:Efn

Field extensions

Template:See The notion of a subfield Template:Math can also be regarded from the opposite point of view, by referring to Template:Mvar being a field extension (or just extension) of Template:Mvar, denoted by

Template:Math,

and read "Template:Mvar over Template:Mvar".

A basic datum of a field extension is its degree Template:Math, i.e., the dimension of Template:Mvar as an Template:Mvar-vector space. It satisfies the formula

Template:Math.

Extensions whose degree is finite are referred to as finite extensions. The extensions Template:Math and Template:Math are of degree Template:Math, whereas Template:Math is an infinite extension.

Algebraic extensions

A pivotal notion in the study of field extensions Template:Math are algebraic elements. An element Template:Math is algebraic over Template:Mvar if it is a root of a polynomial with coefficients in Template:Mvar, that is, if it satisfies a polynomial equation

Template:Math,

with Template:Math in Template:Mvar, and Template:Math. For example, the imaginary unit Template:Mvar in Template:Math is algebraic over Template:Math, and even over Template:Math, since it satisfies the equation

Template:Math.

A field extension in which every element of Template:Mvar is algebraic over Template:Mvar is called an algebraic extension. Any finite extension is necessarily algebraic, as can be deduced from the above multiplicativity formula.

The subfield Template:Math generated by an element Template:Mvar, as above, is an algebraic extension of Template:Mvar if and only if Template:Mvar is an algebraic element. That is to say, if Template:Mvar is algebraic, all other elements of Template:Math are necessarily algebraic as well. Moreover, the degree of the extension Template:Math, i.e., the dimension of Template:Math as an Template:Mvar-vector space, equals the minimal degree Template:Mvar such that there is a polynomial equation involving Template:Mvar, as above. If this degree is Template:Mvar, then the elements of Template:Math have the form

$\sum _{k=0}^{n-1}{a}_k{x}^k,a_k\in E.$

For example, the field Template:Math of Gaussian rationals is the subfield of Template:Math consisting of all numbers of the form Template:Math where both Template:Mvar and Template:Mvar are rational numbers: summands of the form Template:Math (and similarly for higher exponents) do not have to be considered here, since Template:Math can be simplified to Template:Math.

Transcendence bases

The above-mentioned field of rational fractions Template:Math, where Template:Mvar is an indeterminate, is not an algebraic extension of Template:Mvar since there is no polynomial equation with coefficients in Template:Mvar whose zero is Template:Mvar. Elements, such as Template:Mvar, which are not algebraic are called transcendental. Informally speaking, the indeterminate Template:Mvar and its powers do not interact with elements of Template:Mvar. A similar construction can be carried out with a set of indeterminates, instead of just one.

Once again, the field extension Template:Math discussed above is a key example: if Template:Mvar is not algebraic (i.e., Template:Mvar is not a root of a polynomial with coefficients in Template:Mvar), then Template:Math is isomorphic to Template:Math. This isomorphism is obtained by substituting Template:Mvar to Template:Mvar in rational fractions.

A subset Template:Mvar of a field Template:Mvar is a transcendence basis if it is algebraically independent (do not satisfy any polynomial relations) over Template:Mvar and if Template:Mvar is an algebraic extension of Template:Math. Any field extension Template:Math has a transcendence basis. Thus, field extensions can be split into ones of the form Template:Math (purely transcendental extensions) and algebraic extensions.

Closure operations

A field is algebraically closed if it does not have any strictly bigger algebraic extensions or, equivalently, if any polynomial equation

Template:Math, with coefficients Template:Math,

has a solution Template:Math. By the fundamental theorem of algebra, Template:Math is algebraically closed, i.e., any polynomial equation with complex coefficients has a complex solution. The rational and the real numbers are not algebraically closed since the equation

Template:Math

does not have any rational or real solution. A field containing Template:Mvar is called an algebraic closure of Template:Mvar if it is algebraic over Template:Mvar (roughly speaking, not too big compared to Template:Mvar) and is algebraically closed (big enough to contain solutions of all polynomial equations).

By the above, Template:Math is an algebraic closure of Template:Math. The situation that the algebraic closure is a finite extension of the field Template:Mvar is quite special: by the Artin–Schreier theorem, the degree of this extension is necessarily Template:Math, and Template:Mvar is elementarily equivalent to Template:Math. Such fields are also known as real closed fields.

Any field Template:Mvar has an algebraic closure, which is moreover unique up to (non-unique) isomorphism. It is commonly referred to as the algebraic closure and denoted Template:Math. For example, the algebraic closure Template:Math of Template:Math is called the field of algebraic numbers. The field Template:Math is usually rather implicit since its construction requires the ultrafilter lemma, a set-theoretic axiom that is weaker than the axiom of choice. In this regard, the algebraic closure of Template:Math, is exceptionally simple. It is the union of the finite fields containing Template:Math (the ones of order Template:Math). For any algebraically closed field Template:Mvar of characteristic Template:Math, the algebraic closure of the field Template:Math of Laurent series is the field of Puiseux series, obtained by adjoining roots of Template:Mvar.

Fields with additional structure

Since fields are ubiquitous in mathematics and beyond, several refinements of the concept have been adapted to the needs of particular mathematical areas.

Ordered fields

A field F is called an ordered field if any two elements can be compared, so that Template:Math and Template:Math whenever Template:Math and Template:Math. For example, the real numbers form an ordered field, with the usual ordering Template:Math. The Artin–Schreier theorem states that a field can be ordered if and only if it is a formally real field, which means that any quadratic equation

$x_1^2+x_2^2+\dots +x_n^2=0$

only has the solution Template:Math. The set of all possible orders on a fixed field Template:Mvar is isomorphic to the set of ring homomorphisms from the Witt ring Template:Math of quadratic forms over Template:Mvar, to Template:Math.

An Archimedean field is an ordered field such that for each element there exists a finite expression

Template:Math

whose value is greater than that element, that is, there are no infinite elements. Equivalently, the field contains no infinitesimals (elements smaller than all rational numbers); or, yet equivalent, the field is isomorphic to a subfield of Template:Math.

File:Illustration of supremum.svg

Each bounded real set has a least upper bound.

An ordered field is Dedekind-complete if all upper bounds, lower bounds (see Dedekind cut) and limits, which should exist, do exist. More formally, each bounded subset of Template:Mvar is required to have a least upper bound. Any complete field is necessarily Archimedean, since in any non-Archimedean field there is neither a greatest infinitesimal nor a least positive rational, whence the sequence Template:Math, every element of which is greater than every infinitesimal, has no limit.

Since every proper subfield of the reals also contains such gaps, Template:Math is the unique complete ordered field, up to isomorphism. Several foundational results in calculus follow directly from this characterization of the reals.

The hyperreals Template:Math form an ordered field that is not Archimedean. It is an extension of the reals obtained by including infinite and infinitesimal numbers. These are larger, respectively smaller than any real number. The hyperreals form the foundational basis of non-standard analysis.

Topological fields

Another refinement of the notion of a field is a topological field, in which the set Template:Mvar is a topological space, such that all operations of the field (addition, multiplication, the maps Template:Math and Template:Math) are continuous maps with respect to the topology of the space. The topology of all the fields discussed below is induced from a metric, i.e., a function

Template:Math

that measures a distance between any two elements of Template:Mvar.

The completion of Template:Mvar is another field in which, informally speaking, the "gaps" in the original field Template:Mvar are filled, if there are any. For example, any irrational number Template:Mvar, such as Template:Math, is a "gap" in the rationals Template:Math in the sense that it is a real number that can be approximated arbitrarily closely by rational numbers Template:Math, in the sense that distance of Template:Mvar and Template:Math given by the absolute value Template:Math is as small as desired. The following table lists some examples of this construction. The fourth column shows an example of a zero sequence, i.e., a sequence whose limit (for Template:Math) is zero.

Field	Metric	Completion	zero sequence
Template:Math	Template:Math (usual absolute value)	R	Template:Math
Template:Math	obtained using the p-adic valuation, for a prime number Template:Mvar	Template:Math ([[p-adic number|Template:Mvar-adic numbers]])	Template:Math
Template:Math (Template:Mvar any field)	obtained using the Template:Mvar-adic valuation	Template:Math	Template:Math

The field Template:Math is used in number theory and [[p-adic analysis|Template:Mvar-adic analysis]]. The algebraic closure Template:Math carries a unique norm extending the one on Template:Math, but is not complete. The completion of this algebraic closure, however, is algebraically closed. Because of its rough analogy to the complex numbers, it is sometimes called the [[complex p-adic number|complex Template:Mvar-adic numbers]] and is denoted Template:Math.

Local fields

The following topological fields are called local fields:Template:Efn

• finite extensions of Template:Math (local fields of characteristic zero)
• finite extensions of Template:Math, the field of Laurent series over Template:Math (local fields of characteristic Template:Mvar).

These two types of local fields share some fundamental similarities. In this relation, the elements Template:Math and Template:Math (referred to as uniformizer) correspond to each other. The first manifestation of this is at an elementary level: the elements of both fields can be expressed as power series in the uniformizer, with coefficients in Template:Math. (However, since the addition in Template:Math is done using carrying, which is not the case in Template:Math, these fields are not isomorphic.) The following facts show that this superficial similarity goes much deeper:

• Any first-order statement that is true for almost all Template:Math is also true for almost all Template:Math. An application of this is the Ax–Kochen theorem describing zeros of homogeneous polynomials in Template:Math.
• Tamely ramified extensions of both fields are in bijection to one another.
• Adjoining arbitrary Template:Mvar-power roots of Template:Mvar (in Template:Math), respectively of Template:Mvar (in Template:Math), yields (infinite) extensions of these fields known as perfectoid fields. Strikingly, the Galois groups of these two fields are isomorphic, which is the first glimpse of a remarkable parallel between these two fields: $$\mathrm{Gal}\left({\mathbf{𝐐}}_p\right(p^{1/p^{\mathrm{\infty }}}\left)\right)\cong \mathrm{Gal}({\mathbf{𝐅}}_p((t))(t^{1/p^{\mathrm{\infty }}}\left)\right).$$

Differential fields

Differential fields are fields equipped with a derivation, i.e., allow to take derivatives of elements in the field. For example, the field Template:Math, together with the standard derivative of polynomials forms a differential field. These fields are central to differential Galois theory, a variant of Galois theory dealing with linear differential equations.

Galois theory

Galois theory studies algebraic extensions of a field by studying the symmetry in the arithmetic operations of addition and multiplication. An important notion in this area is that of finite Galois extensions Template:Math, which are, by definition, those that are separable and normal. The primitive element theorem shows that finite separable extensions are necessarily simple, i.e., of the form

Template:Math,

where Template:Mvar is an irreducible polynomial (as above). For such an extension, being normal and separable means that all zeros of Template:Mvar are contained in Template:Mvar and that Template:Mvar has only simple zeros. The latter condition is always satisfied if Template:Mvar has characteristic Template:Math.

For a finite Galois extension, the Galois group Template:Math is the group of field automorphisms of Template:Mvar that are trivial on Template:Mvar (i.e., the bijections Template:Math that preserve addition and multiplication and that send elements of Template:Mvar to themselves). The importance of this group stems from the fundamental theorem of Galois theory, which constructs an explicit one-to-one correspondence between the set of subgroups of Template:Math and the set of intermediate extensions of the extension Template:Math. By means of this correspondence, group-theoretic properties translate into facts about fields. For example, if the Galois group of a Galois extension as above is not solvable (cannot be built from abelian groups), then the zeros of Template:Mvar cannot be expressed in terms of addition, multiplication, and radicals, i.e., expressions involving $\sqrt[n]{}$. For example, the symmetric groups Template:Math is not solvable for Template:Math. Consequently, as can be shown, the zeros of the following polynomials are not expressible by sums, products, and radicals. For the latter polynomial, this fact is known as the Abel–Ruffini theorem:

Template:Math (and Template:Math),

Template:Math (where Template:Mvar is regarded as a polynomial in Template:Math, for some indeterminates Template:Math, Template:Mvar is any field, and Template:Math).

The tensor product of fields is not usually a field. For example, a finite extension Template:Math of degree Template:Mvar is a Galois extension if and only if there is an isomorphism of Template:Mvar-algebras

Template:Math.

This fact is the beginning of Grothendieck's Galois theory, a far-reaching extension of Galois theory applicable to algebro-geometric objects.

Invariants of fields

Basic invariants of a field Template:Mvar include the characteristic and the transcendence degree of Template:Mvar over its prime field. The latter is defined as the maximal number of elements in Template:Mvar that are algebraically independent over the prime field. Two algebraically closed fields Template:Mvar and Template:Mvar are isomorphic precisely if these two data agree. This implies that any two uncountable algebraically closed fields of the same cardinality and the same characteristic are isomorphic. For example, Template:Math and Template:Math are isomorphic (but not isomorphic as topological fields).

Model theory of fields

In model theory, a branch of mathematical logic, two fields Template:Mvar and Template:Mvar are called elementarily equivalent if every mathematical statement that is true for Template:Mvar is also true for Template:Mvar and conversely. The mathematical statements in question are required to be first-order sentences (involving Template:Math, Template:Math, the addition and multiplication). A typical example, for Template:Math, Template:Mvar an integer, is

Template:Math = "any polynomial of degree Template:Mvar in Template:Mvar has a zero in Template:Mvar"

The set of such formulas for all Template:Mvar expresses that Template:Mvar is algebraically closed. The Lefschetz principle states that Template:Math is elementarily equivalent to any algebraically closed field Template:Mvar of characteristic zero. Moreover, any fixed statement Template:Math holds in Template:Math if and only if it holds in any algebraically closed field of sufficiently high characteristic.

If Template:Mvar is an ultrafilter on a set Template:Mvar, and Template:Math is a field for every Template:Mvar in Template:Mvar, the ultraproduct of the Template:Math with respect to Template:Mvar is a field. It is denoted by

Template:Math,

since it behaves in several ways as a limit of the fields Template:Math: Łoś's theorem states that any first order statement that holds for all but finitely many Template:Math, also holds for the ultraproduct. Applied to the above sentence Template:Math, this shows that there is an isomorphismTemplate:Efn

${\mathrm{ulim}}_{p\to \mathrm{\infty }}{\overline{\mathbf{𝐅}}}_p\cong \mathbf{𝐂}.$

The Ax–Kochen theorem mentioned above also follows from this and an isomorphism of the ultraproducts (in both cases over all primes Template:Mvar)

Template:Math.

In addition, model theory also studies the logical properties of various other types of fields, such as real closed fields or exponential fields (which are equipped with an exponential function Template:Math).

Absolute Galois group

For fields that are not algebraically closed (or not separably closed), the absolute Galois group Template:Math is fundamentally important: extending the case of finite Galois extensions outlined above, this group governs all finite separable extensions of Template:Mvar. By elementary means, the group Template:Math can be shown to be the Prüfer group, the profinite completion of Template:Math. This statement subsumes the fact that the only algebraic extensions of Template:Math are the fields Template:Math for Template:Math, and that the Galois groups of these finite extensions are given by

Template:Math.

A description in terms of generators and relations is also known for the Galois groups of Template:Mvar-adic number fields (finite extensions of Template:Math).

Representations of Galois groups and of related groups such as the Weil group are fundamental in many branches of arithmetic, such as the Langlands program. The cohomological study of such representations is done using Galois cohomology. For example, the Brauer group, which is classically defined as the group of [[central simple algebra|central simple Template:Mvar-algebras]], can be reinterpreted as a Galois cohomology group, namely

Template:Math.

K-theory

Milnor K-theory is defined as

$K_n^M(F)=F^{\times }\otimes \cdots \otimes F^{\times }/⟨x\otimes (1-x)\mid x\in F\setminus \{0,1\}⟩.$

The norm residue isomorphism theorem, proved around 2000 by Vladimir Voevodsky, relates this to Galois cohomology by means of an isomorphism

$K_n^M(F)/p=H^n(F,{\mu }_l^{\otimes n}).$

Algebraic K-theory is related to the group of invertible matrices with coefficients the given field. For example, the process of taking the determinant of an invertible matrix leads to an isomorphism Template:Math. Matsumoto's theorem shows that Template:Math agrees with Template:Math. In higher degrees, K-theory diverges from Milnor K-theory and remains hard to compute in general.

Applications

Linear algebra and commutative algebra

If Template:Math, then the equation

Template:Math

has a unique solution Template:Mvar in a field Template:Mvar, namely $x=a^{-1}b.$ This immediate consequence of the definition of a field is fundamental in linear algebra. For example, it is an essential ingredient of Gaussian elimination and of the proof that any vector space has a basis.

The theory of modules (the analogue of vector spaces over rings instead of fields) is much more complicated, because the above equation may have several or no solutions. In particular systems of linear equations over a ring are much more difficult to solve than in the case of fields, even in the specially simple case of the ring Template:Math of the integers.

Finite fields: cryptography and coding theory

File:ECClines.svg

The sum of three points Template:Mvar, Template:Mvar, and Template:Mvar on an elliptic curve Template:Mvar (red) is zero if there is a line (blue) passing through these points.

A widely applied cryptographic routine uses the fact that discrete exponentiation, i.e., computing

Template:Math (Template:Mvar factors, for an integer Template:Math)

in a (large) finite field Template:Math can be performed much more efficiently than the discrete logarithm, which is the inverse operation, i.e., determining the solution Template:Mvar to an equation

Template:Math.

In elliptic curve cryptography, the multiplication in a finite field is replaced by the operation of adding points on an elliptic curve, i.e., the solutions of an equation of the form

Template:Math.

Finite fields are also used in coding theory and combinatorics.

Geometry: field of functions

File:Double torus illustration.png

A compact Riemann surface of genus two (two handles). The genus can be read off the field of meromorphic functions on the surface.

Functions on a suitable topological space Template:Mvar into a field Template:Mvar can be added and multiplied pointwise, e.g., the product of two functions is defined by the product of their values within the domain:

Template:Math.

This makes these functions a Template:Mvar-commutative algebra.

For having a field of functions, one must consider algebras of functions that are integral domains. In this case the ratios of two functions, i.e., expressions of the form

$\frac{f(x)}{g(x)},$

form a field, called field of functions.

This occurs in two main cases. When Template:Mvar is a complex manifold Template:Mvar. In this case, one considers the algebra of holomorphic functions, i.e., complex differentiable functions. Their ratios form the field of meromorphic functions on Template:Mvar.

The function field of an algebraic variety Template:Mvar (a geometric object defined as the common zeros of polynomial equations) consists of ratios of regular functions, i.e., ratios of polynomial functions on the variety. The function field of the Template:Mvar-dimensional space over a field Template:Mvar is Template:Math, i.e., the field consisting of ratios of polynomials in Template:Mvar indeterminates. The function field of Template:Mvar is the same as the one of any open dense subvariety. In other words, the function field is insensitive to replacing Template:Mvar by a (slightly) smaller subvariety.

The function field is invariant under isomorphism and birational equivalence of varieties. It is therefore an important tool for the study of abstract algebraic varieties and for the classification of algebraic varieties. For example, the dimension, which equals the transcendence degree of Template:Math, is invariant under birational equivalence. For curves (i.e., the dimension is one), the function field Template:Math is very close to Template:Mvar: if Template:Mvar is smooth and proper (the analogue of being compact), Template:Mvar can be reconstructed, up to isomorphism, from its field of functions.Template:Efn In higher dimension the function field remembers less, but still decisive information about Template:Mvar. The study of function fields and their geometric meaning in higher dimensions is referred to as birational geometry. The minimal model program attempts to identify the simplest (in a certain precise sense) algebraic varieties with a prescribed function field.

Number theory: global fields

Global fields are in the limelight in algebraic number theory and arithmetic geometry. They are, by definition, number fields (finite extensions of Template:Math) or function fields over Template:Math (finite extensions of Template:Math). As for local fields, these two types of fields share several similar features, even though they are of characteristic Template:Math and positive characteristic, respectively. This function field analogy can help to shape mathematical expectations, often first by understanding questions about function fields, and later treating the number field case. The latter is often more difficult. For example, the Riemann hypothesis concerning the zeros of the Riemann zeta function (open as of 2017) can be regarded as being parallel to the Weil conjectures (proven in 1974 by Pierre Deligne).

File:One5Root.svg

The fifth roots of unity form a regular pentagon.

Cyclotomic fields are among the most intensely studied number fields. They are of the form Template:Math, where Template:Math is a primitive Template:Mvarth root of unity, i.e., a complex number Template:Math that satisfies Template:Math and Template:Math for all Template:Math. For Template:Mvar being a regular prime, Kummer used cyclotomic fields to prove Fermat's Last Theorem, which asserts the non-existence of rational nonzero solutions to the equation

Template:Math.

Local fields are completions of global fields. Ostrowski's theorem asserts that the only completions of Template:Math, a global field, are the local fields Template:Math and Template:Math. Studying arithmetic questions in global fields may sometimes be done by looking at the corresponding questions locally. This technique is called the local–global principle. For example, the Hasse–Minkowski theorem reduces the problem of finding rational solutions of quadratic equations to solving these equations in Template:Math and Template:Math, whose solutions can easily be described.

Unlike for local fields, the Galois groups of global fields are not known. Inverse Galois theory studies the (unsolved) problem whether any finite group is the Galois group Template:Math for some number field Template:Mvar. Class field theory describes the abelian extensions, i.e., ones with abelian Galois group, or equivalently the abelianized Galois groups of global fields. A classical statement, the Kronecker–Weber theorem, describes the maximal abelian Template:Math extension of Template:Math: it is the field

Template:Math

obtained by adjoining all primitive Template:Mvarth roots of unity. Kronecker's Jugendtraum asks for a similarly explicit description of Template:Math of general number fields Template:Mvar. For imaginary quadratic fields, $F=\mathbf{𝐐}(\sqrt{-d})$, Template:Math, the theory of complex multiplication describes Template:Math using elliptic curves. For general number fields, no such explicit description is known.

Related notions

In addition to the additional structure that fields may enjoy, fields admit various other related notions. Since in any field Template:Math, any field has at least two elements. Nonetheless, there is a concept of field with one element, which is suggested to be a limit of the finite fields Template:Math, as Template:Mvar tends to Template:Math. In addition to division rings, there are various other weaker algebraic structures related to fields such as quasifields, near-fields and semifields.

There are also proper classes with field structure, which are sometimes called Fields, with a capital 'F'. The surreal numbers form a Field containing the reals, and would be a field except for the fact that they are a proper class, not a set. The nimbers, a concept from game theory, form such a Field as well.

Division rings

Dropping one or several axioms in the definition of a field leads to other algebraic structures. As was mentioned above, commutative rings satisfy all field axioms except for the existence of multiplicative inverses. Dropping instead commutativity of multiplication leads to the concept of a division ring or skew field; sometimes associativity is weakened as well. Historically, division rings were sometimes referred to as fields, while fields were called "commutative fields". The only division rings that are finite-dimensional Template:Math-vector spaces are Template:Math itself, Template:Math (which is a field), and the quaternions Template:Math (in which multiplication is non-commutative). This result is known as the Frobenius theorem. The octonions Template:Math, for which multiplication is neither commutative nor associative, is a normed alternative division algebra, but is not a division ring. This fact was proved using methods of algebraic topology in 1958 by Michel Kervaire, Raoul Bott, and John Milnor.

Wedderburn's little theorem states that all finite division rings are fields.

References

• , especially Chapter 13
• . See especially Book 3 (Template:ISBN) and Book 6 (Template:ISBN).
• Template:Springer