Definitions

distributive-property

Field (mathematics)

In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.

All fields are rings, but not conversely. Fields differ from rings most importantly in the requirement that division be possible, but also, in modern definitions, by the requirement that the multiplication operation in a field be commutative. Otherwise the structure is a so-called skew field (better known as a division ring), although historically division rings were called fields and fields were commutative fields.

The prototypical example of a field is Q, the field of rational numbers. Other important examples include the field of real numbers R, the field of complex numbers C and, for any prime number p, the finite field of integers modulo p, denoted Z/pZ, Fp or GF(p). For any field K, the set K(X) of rational functions with coefficients in K is also a field.

The mathematical discipline concerned with the study of fields is called field theory.

A field is a specific type of integral domain, and can be characterized by the following (not necessarily exhaustive) chain of class inclusions:

Definition and illustration

A starting point for the notion of a field is the set Q of rational numbers. In Q, there are four essential operations: addition together with subtraction, and multiplication with division. Intuitively, a field is a set of numbers which has four such operations. In order to qualify as a field, these operations have to satisfy certain axioms.

A field is a set together with two operations, usually called addition and multiplication, and denoted "+" and "·", respectively, such that the following axioms hold (In the following, a, b, and c are any three elements of the field F.)

  1. Closure of F under + and · : For all a, b belonging to F, both a + b and a · b belong to F (or more formally, + and · are binary operations on F).
  2. Associativity of addition and multiplication: The following identities hold: a + (b + c) = (a + b) + c and a · (b · c) = (a · b) · c.
  3. Commutativity of addition and multiplication: a + b = b + a and a · b = b · a.
  4. Additive and multiplicative identity: There exists an element 0 ∈ F, called additive identity element, such that for all a, a + 0 = a. Likewise there is an element 1, the multiplicative identity element, such that for all a belonging to F, a · 1 = a. For technical reasons, 1 is required not to equal 0.
  5. Additive and multiplicative inverses: For every a belonging to F, there exists an element −a in F, such that a + (−a) = 0. Similarly, for any nonzero a, i.e. for any a ≠ 0, there exists an element a−1 in F, such that a · a−1 = 1.
  6. The distributivity axiom is the axiom which specifies how addition and multiplication intertwine: for all a, b, c, the equality a · (b + c) = (a · b) + (a · c) is required to hold.

First example: rational numbers

The easiest example for a field are the rational numbers consisting of 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, as well—is b/a. To see the latter note that

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

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

frac{a}{b} cdot left(frac{c}{d} + frac{e}{f}right) = 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}text{,}
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 rôle of the additive identity element (denoted 0 in the axioms), and I is the multiplicative identity (denoted 1 above). Checking that all field axioms are indeed satisfied is easy, if tedious. For example:
A · (B + A) = A · 1 = 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, denoted F4. Field theory is concerned with understanding the reasons for the existence of this field, defined in a fairly ad-hoc manner, and with 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. Indeed A = A1, B = A2 = A · A, and finally I = A3 = A · A · A. This is no coincidence, but one of the starting points of a deeper understanding of (finite) fields.

Related algebraic structures

Ring and field axioms
Abelian group Ring Commutative
ring
Skew field or
Division ring
Field
Abelian (additive) group
structure
Multiplicative structure
and distributivity
Commutativity of multiplication
Multiplicative inverses
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 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 , 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 5 or higher.

In 1871, Richard Dedekind called a set of real or complex numbers which is closed under the four arithmetic operations a "field". He used the German word Körper – "body" for this notion, 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.

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. In 1910 Ernst Steinitz published the very influential paper Algebraische Theorie der Körper (German: Algebraic Theory of Fields). 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 during 1928-1942.

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 Q3] consisting of expressions
a + b · ζ + c · ζ2, a, b, cQ
where ζ is a third root of unity, i.e. a complex number satisfying ζ3 = 1, , 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, Q3] is a field extension of degree 3. 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 acbd + (bc + ad)i.

The real numbers can be constructed by completing the rational numbers, i.e. filling the "gaps": for example √p-adic numbers Qp 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 to construct 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 to settle these problems. 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 fF, the square root of f is also a constructible number.

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. Highlighted in the multiplication and addition tables above is the field F2 consisting of two elements O and I. 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 Logical AND operation, this field finds applications in computer science, especially in cryptography and coding theory.

In any 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 1 + 1 + ... + 1 is never zero, for any number of summands, such as in Q, for example, the characteristic is said to be zero.)

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

Fp = Z/pZ = {0, 1, ..., p − 1},
where the operations are defined by performing the operation in Z, dividing by p and taking the remainder; see modular arithmetic. Any field of characteristic p necessarily contains Fp, and thus can be viewed as a vector space over the smaller field, of finite dimension. I.e. any finite field has to have q = pn elements, n > 0. By developing more field theory, in particular the notion of the splitting field, which is, in general a field containing all roots of a given polynomial, and minimal with this property, one can show that any two finite fields with the same number of elements have indeed the same field structure, i.e. multiplication and addition behave the same way. The finite field with q elements is usually denoted Fq.

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, due to the presence of possible zeros, i.e. points xX where f(x) = 0, one has to take poles into account, i.e. formally allowing f(x) = ∞.

If X is an algebraic variety over F, then the rational functions VF, i.e. functions defined almost everywhere, form a field, the function field of V. Likewise, if X is a Riemann surface, then the meromorphic functions SC 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. Morally, local fields are completions of global fields at a given place. For example, Q is a global fields, and the attached local fields are Qp and R (Ostrowski's theorem). Algebraic number fields and function fields over Fq 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 Fq×, it is cyclic of order . In the introductory example, a generator of F4× is the element A.
  • From the point of view of algebraic geometry, fields are points, because the spectrum Spec F has only one point, corresponding to the 0-ideal. This entails that a commutative ring is a field if and only if it has no ideals except {0} and itself. Equivalently, an integral domain is 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 , called the algebraic closure of F, which contains F, is algebraic over F, which means that any element x of satisfies a polynomial equation
fnxn + fn−1xn−1 + ... + f1x + f0 = 0, with coefficients fn, ..., f0F,
and is algebraically closed, i.e. any such polynomial does have at least one solution in . 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 as being uniquely determined by F, since the isomorphism above is not itself unique. In these cases, one refers to such a 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 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.

The notion of field extension lies at the heart of field theory, and as such is crucial to many other algebraic domains. A field extension F / E is simply a field F and a subfield EF. 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. expressions of the kind

frac{p(X)}{q(X)}, 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 F. If one allows formal power series (also called Laurent series) in both denominator and numerator, one also gets a field, denoted E((X)).

In the above two cases, one is just adding a new symbol, namely X, which does not interact with elements of E. The following construction is different in this respect. This idea will first be exemplified by R vs. C. As explained above, C is a extension of R. The essential new element of C, in comparison to R is the imaginary unit i. It satisfies i2 = −1, or equivalently

i2 + 1 = 0.
Yet equivalently phrased, i is a zero of the polynomial p(X) = X2 + 1. For any field F, the ring of polynomials with coefficients in F is denoted by F[X]. The corresponding quotient C = R[X] / (X2 + 1) contains all real numbers and a variable X. In C, however, the relation X2 + 1 = 0 holds. In other words, the element XC satisfies exactly the property that i does. Therefore, the abstractly constructed C is isomorphic to the field C of complex numbers.

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 F = E[X] / (p(X)), where (p(X)) denotes the ideal generated by 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 E not in G. 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. 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 Z is just Q. Also, the field F(X) is the quotient field of the ring of polynomials F[X]. Getting back the ring from the field is sometimes possible; see discrete valuation ring.

Another method to obtain a field from a commutative ring R is taking the quotient , 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 Fp = Z / pZ.

Ultraproducts

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

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 : FF 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, Esep, of E over E (i.e. Gal(Esep/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. 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 Esep/E it states that there this a one-to-one correspondence between closed subgroups of Gal(Esep/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 Esep, 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 set of all surreal numbers with birthday smaller than some inaccessible cardinal form a field.
  • 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.

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 and coding theory, and again algebraic extension is an important tool.

Binary fields, fields of characteristic 2, are useful in computer science.

See also

Notes

References

  • , especially Chapter 13
  • . See especially Book 3 (ISBN 0-521-27288-2) and Book 6 (ISBN 0-521-27291-2).

External links

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