Field (mathematics)

Algebraic structures
Group-like Semigroup / Monoid Racks and quandles Quasigroup and loop Abelian group Magma Lie group Group theory
Ring-like Ring Rng Semiring Near-ring Commutative ring Domain Integral domain Field Ring theory
Lattice-like Lattice Semilattice Map of lattices Lattice theory
Module-like Module Group with operators Vector space
Algebra-like Algebra Associative Non-associative Lie algebra Graded Bialgebra

In mathematics, a field is one of the fundamental algebraic structures used in abstract algebra. It is a nonzero commutative division ring, or equivalently a ring whose nonzero elements form an abelian group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division satisfying the appropriate abelian group equations and distributive law. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, algebraic function fields, algebraic number fields, p-adic fields, and so forth.

Any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. The theory of field extensions (including Galois theory) involves the roots of polynomials with coefficients in a field; among other results, this theory leads to impossibility proofs for the classical problems of angle trisection and squaring the circle with a compass and straightedge, as well as a proof of the Abel–Ruffini theorem on the algebraic insolubility of quintic equations. In modern mathematics, the theory of fields (or field theory) plays an essential role in number theory and algebraic geometry.

As an algebraic structure, every field is a ring, but not every ring is a field. The most important difference is that fields allow for division (though not division by zero), while a ring need not possess multiplicative inverses; for example the integers form a ring, but 2x = 1 has no solution in integers. Also, the multiplication operation in a field is required to be commutative. A ring in which division is possible but commutativity is not assumed (such as the quaternions) is called a division ring or skew field. (Historically, division rings were sometimes referred to as fields, while fields were called commutative fields.)

As a ring, a field may be classified as a specific type of integral domain, and can be characterized by the following (not exhaustive) chain of class inclusions:

commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields

Definition and illustration

Intuitively, a field is a set F that is a commutative group with respect to two compatible operations, addition and multiplication (the latter excluding zero), with "compatible" being formalized by distributivity, and the caveat that the additive and the multiplicative identities are distinct (0 ≠ 1).

The most common way to formalize this is by defining a field as a set together with two operations, usually called addition and multiplication, and denoted by + and ·, respectively, such that the following axioms hold (note that subtraction and division are defined in terms of the inverse operations of addition and multiplication):^{[note 1]}

Closure of F under addition and multiplication

For all a, b in F, both a + b and a · b are in F (or more formally, + and · are binary operations on F).

Associativity of addition and multiplication

For all a, b, and c in F, the following equalities hold: a + (b + c) = (a + b) + c and a · (b · c) = (a · b) · c.

Commutativity of addition and multiplication

For all a and b in F, the following equalities hold: a + b = b + a and a · b = b · a.

Existence of additive and multiplicative identity elements

There exists an element of F, called the additive identity element and denoted by 0, such that for all a in F, a + 0 = a. Likewise, there is an element, called the multiplicative identity element and denoted by 1, such that for all a in F, a · 1 = a. To exclude the trivial ring, the additive identity and the multiplicative identity are required to be distinct.

Existence of additive inverses and multiplicative inverses

For every a in F, there exists an element −a in F, such that a + (−a) = 0. Similarly, for any a in F other than 0, there exists an element a⁻¹ in F, such that a · a⁻¹ = 1. (The elements a + (−b) and a · b⁻¹ are also denoted a − b and a/b, respectively.) In other words, subtraction and division operations exist.

Distributivity of multiplication over addition

For all a, b and c in F, the following equality holds: a · (b + c) = (a · b) + (a · c).

A field is therefore an algebraic structure 〈F, +, ·, −, ⁻¹, 0, 1〉; of type 〈2, 2, 1, 1, 0, 0〉, consisting of two abelian groups:

• F under +, −, and 0;
• F ∖ {0} under ·, ⁻¹, and 1, with 0 ≠ 1,

with · distributing over +.^[1]

First example: rational numbers

A simple example of a field is the field of rational numbers, consisting of numbers which can be written as fractions a/b, where a and b are integers, and b ≠ 0. The additive inverse of such a fraction is simply −a/b, and the multiplicative inverse (provided that a ≠ 0) is b/a. To see the latter, note that

The abstractly required field axioms reduce to standard properties of rational numbers, such as the law of distributivity

or the law of commutativity and law of associativity.

Second example: a field with four elements

+	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), and I is the multiplicative identity (denoted 1 above). One can check that all field axioms are satisfied. For example:

A · (B + A) = A · I = A, which equals A · B + A · A = I + B = A, as required by the distributivity.

The above field is called a finite field with four elements, and can be denoted F₄. Field theory is concerned with understanding the reasons for the existence of this field, defined in a fairly ad-hoc manner, and describing its inner structure. For example, from a glance at the multiplication table, it can be seen that any non-zero element (i.e., I, A, and B) is a power of A: A = A¹, B = A² = A · A, and finally I = A³ = A · A · A. This is not a coincidence, but rather one of the starting points of a deeper understanding of (finite) fields.

Alternative axiomatizations

As with other algebraic structures, there exist alternative axiomatizations. Because of the relations between the operations, one can alternatively axiomatize a field by explicitly assuming that there are four binary operations (add, subtract, multiply, divide) with axioms relating these, or (by functional decomposition) in terms of two binary operations (add and multiply) and two unary operations (additive inverse and multiplicative inverse), or other variants.

The usual axiomatization in terms of the two operations of addition and multiplication is brief and allows the other operations to be defined in terms of these basic ones, but in other contexts, such as topology and category theory, it is important to include all operations as explicitly given, rather than implicitly defined (compare topological group). This is because without further assumptions, the implicitly defined inverses may not be continuous (in topology), or may not be able to be defined (in category theory). Defining an inverse requires that one is working with a set, not a more general object.

For a very economical axiomatization of the field of real numbers, whose primitives are merely a set R with 1 ∈ R, addition, and a binary relation, "<". See Tarski's axiomatization of the reals.

Related algebraic structures

Ring and field axioms
	Ring	Commutative ring	Skew field or Division ring	Field
Abelian (additive) group structure	Yes	Yes	Yes	Yes
Multiplicative structure and distributivity	Yes	Yes	Yes	Yes
Commutativity of multiplication	No	Yes	No	Yes
Multiplicative inverses	No	No	Yes	Yes

The axioms imposed above resemble the ones familiar from other algebraic structures. For example, the existence of the binary operation "·", together with its commutativity, associativity, (multiplicative) identity element and inverses are precisely the axioms for an abelian group. In other words, for any field, the subset of nonzero elements F \ {0}, also often denoted F^×, is an abelian group (F^×, ·) usually called multiplicative group of the field. Likewise (F, +) is an abelian group. The structure of a field is hence the same as specifying such two group structures (on the same set), obeying the distributivity.

Important other algebraic structures such as rings arise when requiring only part of the above axioms. For example, if the requirement of commutativity of the multiplication operation · is dropped, one gets structures usually called division rings or skew fields.

Remarks

By elementary group theory, applied to the abelian groups (F^×, ·), and (F, +), the additive inverse −a and the multiplicative inverse a⁻¹ are uniquely determined by a.

Similar direct consequences from the field axioms include

−(a · b) = (−a) · b = a · (−b), in particular −a = (−1) · a

as well as

a · 0 = 0.

Both can be shown by replacing b or c with 0 in the distributive property.

History

The concept of field was used implicitly by Niels Henrik Abel and Évariste Galois in their work on the solvability of polynomial equations with rational coefficients of degree five or higher.

In 1857, Karl von Staudt published his Algebra of Throws which provided a geometric model satisfying the axioms of a field.^[2] This construction has been frequently recalled as a contribution to the foundations of mathematics.

In 1871, Richard Dedekind introduced, for a set of real or complex numbers which is closed under the four arithmetic operations, the German word Körper, which means "body" or "corpus" (to suggest an organically closed entity),^[3] hence the common use of the letter K to denote a field. He also defined rings (then called order or order-modul), but the term "a ring" (Zahlring) was invented by Hilbert.^[4] In 1893, Eliakim Hastings Moore called the concept "field" in English.^[5][6]

In 1881, Leopold Kronecker defined what he called a "domain of rationality", which is indeed a field of polynomials in modern terms. In 1893, Heinrich M. Weber gave the first clear definition of an abstract field.^[7] In 1910, Ernst Steinitz published the very influential paper Algebraische Theorie der Körper (English: Algebraic Theory of Fields).^[8] In this paper he axiomatically studies the properties of fields and defines many important field theoretic concepts like prime field, perfect field and the transcendence degree of a field extension.

Emil Artin developed the relationship between groups and fields in great detail from 1928 through 1942.

Examples

Rationals and algebraic numbers

The field of rational numbers Q has been introduced above. A related class of fields very important in number theory are algebraic number fields. We will first give an example, namely the field Q(ζ) consisting of numbers of the form

a + bζ

with a, b ∈ Q, where ζ is a primitive third root of unity, i.e., a complex number satisfying ζ³ = 1, ζ ≠ 1. This field extension can be used to prove a special case of Fermat's last theorem, which asserts the non-existence of rational nonzero solutions to the equation

x³ + y³ = z³.

In the language of field extensions detailed below, Q(ζ) is a field extension of degree 2. Algebraic number fields are by definition finite field extensions of Q, that is, fields containing Q having finite dimension as a Q-vector space.

Reals, complex numbers, and p-adic numbers

Take the real numbers R, under the usual operations of addition and multiplication. When the real numbers are given the usual ordering, they form a complete ordered field; it is this structure which provides the foundation for most formal treatments of calculus.

The complex numbers C consist of expressions

a + bi

where i is the imaginary unit, i.e., a (non-real) number satisfying i² = −1. Addition and multiplication of real numbers are defined in such a way that all field axioms hold for C. For example, the distributive law enforces

(a + bi)·(c + di) = ac + bci + adi + bdi², which equals ac−bd + (bc + ad)i.

The real numbers can be constructed by completing the rational numbers, i.e., filling the "gaps": for example √2 is such a gap. By a formally very similar procedure, another important class of fields, the field of p-adic numbers Q_p is built. It is used in number theory and p-adic analysis.

Hyperreal numbers and superreal numbers extend the real numbers with the addition of infinitesimal and infinite numbers.

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. Using the field notion and field theory allows these problems to be settled. To do so, the field of constructible numbers is considered. It contains, on the plane, the points 0 and 1, and all complex numbers that can be constructed from these two by a finite number of construction steps using only compass and straightedge. This set, endowed with the usual addition and multiplication of complex numbers does form a field. For example, multiplying two (real) numbers r₁ and r₂ that have already been constructed can be done using construction at the right, based on the intercept theorem. This way, the obtained field F contains all rational numbers, but is bigger than Q, because for any f ∈ F, the square root of f is also a constructible number.

A closely related concept is that of a Euclidean field, namely an ordered field whose positive elements are closed under square root. The real constructible numbers form the least Euclidean field, and the Euclidean fields are precisely the ordered extensions thereof.

Finite fields

Finite fields (also called Galois fields) are fields with finitely many elements. The above introductory example F₄ is a field with four elements. F₂ consists of two elements, 0 and 1. This is the smallest field, because by definition a field has at least two distinct elements 1 ≠ 0. Interpreting the addition and multiplication in this latter field as XOR and AND operations, this field finds applications in computer science, especially in cryptography and coding theory.

In a finite field there is necessarily an integer n such that 1 + 1 + ··· + 1 (n repeated terms) equals 0. It can be shown that the smallest such n must be a prime number, called the characteristic of the field. If a (necessarily infinite) field has the property that 1 + 1 + ··· + 1 is never zero, for any number of summands, such as in Q, for example, the characteristic is said to be zero.

A basic class of finite fields are the fields F_p with p elements (p a prime number):

F_p = Z/pZ = {0, 1, ..., p − 1},

where the operations are defined by performing the operation in the set of integers Z, dividing by p and taking the remainder; see modular arithmetic. A field K of characteristic p necessarily contains F_p,^[9] and therefore may be viewed as a vector space over F_p, of finite dimension if K is finite. Thus a finite field K has prime power order, i.e., K has q = pⁿ elements (where n > 0 is the number of elements in a basis of K over F_p). By developing more field theory, in particular the notion of the splitting field of a polynomial f over a field K, which is the smallest field containing K and all roots of f, one can show that two finite fields with the same number of elements are isomorphic, i.e., there is a one-to-one mapping of one field onto the other that preserves multiplication and addition. Thus we may speak of the finite field with q elements, usually denoted by F_q or GF(q).

Archimedean fields

An Archimedean field is an ordered field such that for each element there exists a finite expression 1 + 1 + ··· + 1 whose value is greater than that element, that is, there are no infinite elements. Equivalently, the field contains no infinitesimals; or, the field is isomorphic to a subfield of the reals. A necessary condition for an ordered field to be complete is that it be Archimedean, since in any non-Archimedean field there is neither a greatest infinitesimal nor a least positive rational, whence the sequence 1/2, 1/3, 1/4, …, every element of which is greater than every infinitesimal, has no limit. (And since every proper subfield of the reals also contains such gaps, up to isomorphism the reals form the unique complete ordered field.)

Field of functions

Given a geometric object X, one can consider functions on such objects. Adding and multiplying them pointwise, i.e., (f ⋅ g)(x) = f(x) ⋅ g(x) this leads to a field. However, for having multiplicative inverses, one has to consider partial functions, which, almost everywhere, are defined and have a non-zero value.

If X is an algebraic variety over a field F, then the rational functions X → F form a field, the function field of X. This field consists of the functions that are defined and are the quotient of two polynomial functions outside some subvariety. Likewise, if S is a Riemann surface, then the meromorphic functions S → C form a field. Under certain circumstances, namely when S is compact, S can be reconstructed from this field.

Local and global fields

Another important distinction in the realm of fields, especially with regard to number theory, are local fields and global fields. Local fields are completions of global fields at a given place. For example, Q is a global field, and the attached local fields are Q_p and R (Ostrowski's theorem). Algebraic number fields and function fields over F_q are further global fields. Studying arithmetic questions in global fields may sometimes be done by looking at the corresponding questions locally—this technique is called local-global principle.

Some first theorems

• Every finite subgroup of the multiplicative group F^× is cyclic. This applies in particular to F_q^×, it is cyclic of order q − 1. In the introductory example, a generator of F₄^× is the element A.
• A integral domain is a field if and only if it has no ideals except {0} and itself. Equivalently, an integral domain is a field if and only if its Krull dimension is 0.
• Isomorphism extension theorem

Constructing fields

Closure operations

Assuming the axiom of choice, for every field F, there exists a field F, called the algebraic closure of F, which contains F, is algebraic over F, which means that any element x of F satisfies a polynomial equation

f_nxⁿ + f_n−1xⁿ⁻¹ + ··· + f₁x + f₀ = 0, with coefficients f_n, ..., f₀ ∈ F,

and is algebraically closed, i.e., any such polynomial does have at least one solution in F. The algebraic closure is unique up to isomorphism inducing the identity on F. However, in many circumstances in mathematics, it is not appropriate to treat F as being uniquely determined by F, since the isomorphism above is not itself unique. In these cases, one refers to such a F as an algebraic closure of F. A similar concept is the separable closure, containing all roots of separable polynomials, instead of all polynomials.

For example, if F = Q, the algebraic closure Q is also called field of algebraic numbers. The field of algebraic numbers is an example of an algebraically closed field of characteristic zero; as such it satisfies the same first-order sentences as the field of complex numbers C.

In general, all algebraic closures of a field are isomorphic. However, there is in general no preferable isomorphism between two closures. Likewise for separable closures.

Subfields and field extensions

A subfield is, informally, a small field contained in a bigger one. Formally, a subfield E of a field F is a subset containing 0 and 1, closed under the operations +, −, · and multiplicative inverses and with its own operations defined by restriction. For example, the real numbers contain several interesting subfields: the real algebraic numbers, the computable numbers and the rational numbers are examples.

The notion of field extension lies at the heart of field theory, and is crucial to many other algebraic domains. A field extension F / E is simply a field F and a subfield E ⊂ F. Constructing such a field extension F / E can be done by "adding new elements" or adjoining elements to the field E. For example, given a field E, the set F = E(X) of rational functions, i.e., equivalence classes of expressions of the kind

where p(X) and q(X) are polynomials with coefficients in E, and q is not the zero polynomial, forms a field. This is the simplest example of a transcendental extension of E. It also is an example of a domain (the ring of polynomials in this case) being embedded into its field of fractions .

The ring of formal power series is also a domain, and again the (equivalence classes of) fractions of the form p(X)/ q(X) where p and q are elements of form the field of fractions for . This field is actually the ring of Laurent series over the field E, denoted .

In the above two cases, the added symbol X and its powers did not interact with elements of E. It is possible however that the adjoined symbol may interact with E. This idea will be illustrated by adjoining an element to the field of real numbers R. As explained above, C is an extension of R. C can be obtained from R by adjoining the imaginary symbol i which satisfies i² = −1. The result is that R[i]=C. This is different from adjoining the symbol X to R, because in that case, the powers of X are all distinct objects, but here, i²=−1 is actually an element of R.

Another way to view this last example is to note that i is a zero of the polynomial p(X) = X² + 1. The quotient ring can be mapped onto C using the map . Since the ideal (X²+1) is generated by a polynomial irreducible over R, the ideal is maximal, hence the quotient ring is a field. This nonzero ring map from the quotient to C is necessarily an isomorphism of rings.

The above construction generalises to any irreducible polynomial in the polynomial ring E[X], i.e., a polynomial p(X) that cannot be written as a product of non-constant polynomials. The quotient ring F = E[X] / (p(X)), is again a field.

Alternatively, constructing such field extensions can also be done, if a bigger container is already given. Suppose given a field E, and a field G containing E as a subfield, for example G could be the algebraic closure of E. Let x be an element of G not in E. Then there is a smallest subfield of G containing E and x, denoted F = E(x) and called field extension F / E generated by x in G.^[10] Such extensions are also called simple extensions. Many extensions are of this type; see the primitive element theorem. For instance, Q(i) is the subfield of C consisting of all numbers of the form a + bi where both a and b are rational numbers.

One distinguishes between extensions having various qualities. For example, an extension K of a field k is called algebraic, if every element of K is a root of some polynomial with coefficients in k. Otherwise, the extension is called transcendental. The aim of Galois theory is the study of algebraic extensions of a field.

Rings vs fields

Adding multiplicative inverses to an integral domain R yields the field of fractions of R. For example, the field of fractions of the integers Z is just Q. Also, the field F(X) is the quotient field of the ring of polynomials F[X].

Another method to obtain a field from a commutative ring R is taking the quotient R / m, where m is any maximal ideal of R. The above construction of F = E[X] / (p(X)), is an example, because the irreducibility of the polynomial p(X) is equivalent to the maximality of the ideal generated by this polynomial. Another example are the finite fields F_p = Z / pZ.

Ultraproducts

If I is an index set, U is an ultrafilter on I, and F_i is a field for every i in I, the ultraproduct of the F_i with respect to U is a field.

For example, a non-principal ultraproduct of finite fields is a pseudo finite field; i.e., a PAC field having exactly one extension of any degree.

Galois theory

Galois theory aims to study the algebraic extensions of a field by studying the symmetry in the arithmetic operations of addition and multiplication. The fundamental theorem of Galois theory shows that there is a strong relation between the structure of the symmetry group and the set of algebraic extensions.

In the case where F / E is a finite (Galois) extension, Galois theory studies the algebraic extensions of E that are subfields of F. Such fields are called intermediate extensions. Specifically, the Galois group of F over E, denoted Gal(F/E), is the group of field automorphisms of F that are trivial on E (i.e., the bijections σ : F → F that preserve addition and multiplication and that send elements of E to themselves), and the fundamental theorem of Galois theory states that there is a one-to-one correspondence between subgroups of Gal(F/E) and the set of intermediate extensions of the extension F/E. The theorem, in fact, gives an explicit correspondence and further properties.

To study all (separable) algebraic extensions of E at once, one must consider the absolute Galois group of E, defined as the Galois group of the separable closure, E^sep, of E over E i.e., Gal(E^sep/E). It is possible that the degree of this extension is infinite (as in the case of E = Q). It is thus necessary to have a notion of Galois group for an infinite algebraic extension. The Galois group in this case is obtained as a "limit" (specifically an inverse limit) of the Galois groups of the finite Galois extensions of E. In this way, it acquires a topology.^{[note 2]} The fundamental theorem of Galois theory can be generalized to the case of infinite Galois extensions by taking into consideration the topology of the Galois group, and in the case of E^sep/E it states that there this a one-to-one correspondence between closed subgroups of Gal(E^sep/E) and the set of all separable algebraic extensions of E (technically, one only obtains those separable algebraic extensions of E that occur as subfields of the chosen separable closure E^sep, but since all separable closures of E are isomorphic, choosing a different separable closure would give the same Galois group and thus an "equivalent" set of algebraic extensions).

Generalizations

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 form a Field. The set of nimbers with birthday smaller than 2²ⁿ, the nimbers with birthday smaller than any infinite cardinal are all examples of fields.

In a different direction, differential fields are fields equipped with a derivation. For example, the field R(X), together with the standard derivative of polynomials forms a differential field. These fields are central to differential Galois theory. Exponential fields, meanwhile, are fields equipped with an exponential function that provides a homomorphism between the additive and multiplicative groups within the field. The usual exponential function makes the real and complex numbers exponential fields, denoted R_exp and C_exp respectively.

Generalizing in a more categorical direction yields the field with one element and related objects.

Exponentiation

One does not in general study generalizations of fields with three binary operations. The familiar addition/subtraction, multiplication/division, exponentiation/root-extraction/logarithm operations from the natural numbers to the reals, each built up in terms of iteration of the last, mean that generalizing exponentiation as a binary operation is tempting, but has generally not proven fruitful; instead, an exponential field assumes a unary exponential function from the additive group to the multiplicative group, not a partially defined binary function. Note that the exponential operation of is neither associative nor commutative, nor has a unique inverse ( are both square roots of 4, for instance), unlike addition and multiplication, and further is not defined for many pairs—for example, does not define a single number. These all show that even for rational numbers exponentiation is not nearly as well-behaved as addition and multiplication, which is why one does not in general axiomatize exponentiation.

Applications

The concept of a field is of use, for example, in defining vectors and matrices, two structures in linear algebra whose components can be elements of an arbitrary field.

Finite fields are used in number theory, Galois theory, cryptography, coding theory and combinatorics; and again the notion of algebraic extension is an important tool.

See also

Notes

References

Sources

External links

The Wikibook Abstract algebra has a page on the topic of: Fields

• Hazewinkel, Michiel, ed. (2001), "Field", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
• Field Theory Q&A
• Fields at ProvenMath definition and basic properties.
• Field at PlanetMath.org.