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abstract algebra, a Oh when the saints...go marching in...?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, F_p 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.

Equivalent definitions

Definition 1

A field is a commutative division ring.

Definition 2

A field is a commutative ring (F, +, *) such that 0 does not equal 1 and all elements of F except 0 have a multiplicative inverse. (Note that 0 and 1 here stand for the identity elements for the + and * operations respectively, which may differ from the familiar real numbers 0 and 1.)

Definition 3

Explicitly, a field is defined by these properties:

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).
Both + and * are associative: For all a, b, c in F, a + (b + c) = (a + b) + c and a * (b * c) = (a * b) * c.
Both + and * are commutative: For all a, b belonging to F, a + b = b + a and a * b = b * a.
The operation * is distributive over the operation +: For all a, b, c, belonging to F, a * (b + c) = (a * b) + (a * c).
Existence of an additive identity: There exists an element 0 in F, such that for all a belonging to F, a + 0 = a.
Existence of a multiplicative identity: There exists an element 1 in F different from 0, such that for all a belonging to F, a * 1 = a.
Existence of additive inverses: For every a belonging to F, there exists an element −a in F, such that a + (−a) = 0.
Existence of multiplicative inverses: For every a ≠ 0 belonging to F, there exists an element a⁻¹ in F, such that a * a⁻¹ = 1.

The requirement 0 ≠ 1 ensures that the set which only contains a single element is not a field. Directly from the axioms, one may show that (F, +) and (F \ {0}, *) are commutative groups (abelian groups) and that therefore (see elementary group theory) the additive inverse −a and the multiplicative inverse a⁻¹ are uniquely determined by a. Other useful rules include

−a = (−1) * a

and more generally

−(a * b) = (−a) * b = a * (−b)

as well as

a * 0 = 0,

all rules familiar from elementary arithmetic.

If the requirement of commutativity of the operation * is dropped, one distinguishes the above commutative fields from non-commutative fields. Fields which are not assumed to be commutative are usually called division rings or skew fields.

History

The concept of a field is due to Dedekind, who used the word Körper "body" for this notion. He also was the first to define rings (then called order or order-modul), but the term "a ring" (Zahlring) was invented by Hilbert. ^[1]

Examples

The complex numbers C, under the usual operations of addition and multiplication. The field of complex numbers contains the following subfields (a subfield of a field F is a set containing 0 and 1, closed under the operations + , - and * of F and with its own operations defined by restriction):
The rational numbers Q = { a/b | a, b in Z, b ≠ 0 } where Z is the set of integers. The field of rational numbers contains no proper subfields.
- An algebraic number field is a finite field extension of the rational numbers Q, that is, a field containing Q which has finite dimension as a vector space over Q. Any such field is isomorphic to a subfield of C, and any such isomorphism induces the identity on Q. These fields are very important in number theory.
- The field of algebraic numbers ${\overline {\mathbb {Q} }}$ , the algebraic closure of Q. 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.
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 real numbers contain several interesting subfields: the real algebraic numbers, the computable numbers.
There is (up to isomorphism) exactly one finite field with q elements, for every finite number q which is a power of a prime number, q≠ 1. (No finite field can exist with any other number of elements.) This is usually denoted F_q . Finite fields are also called Galois fields.
- In particular, for a given prime number p, the set of integers modulo p is a finite field with p elements: Z/pZ = F_p = {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.
- Taking p = 2, we obtain the smallest field, F₂, which has only two elements: 0 and 1. It can be defined by the two Cayley tables

     +  0  1        *  0  1
     0  0  1        0  0  0
     1  1  0        1  0  1

This field has important uses in computer science, especially in cryptography and coding theory.

The rational numbers can be extended to the fields of p-adic numbers for every prime number p. These fields are important in both number theory and mathematical analysis.
Let E and F be two fields with F a subfield of E. Let x be an element of E not in F. Then there is a smallest subfield of E containing F and x, denoted F(x). We call F(x) a simple extension of F. 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. In fact, it can be shown that every number field is a simple extension of Q.
For a given field F, the set F(X) of rational functions in the variable X with coefficients in F is a field; this is the quotient field of the ring of polynomials F[X]. This is the simplest example of a transcendental extension of F.
If F is a field, and p(X) is an irreducible polynomial in the polynomial ring F[X], then the quotient F[X]/<p(X)> , where <p(X)> denotes the ideal generated by p(X), is a field with a subfield isomorphic to F. For instance, R[X]/<X² + 1> is a field (in fact, it is isomorphic to the field of complex numbers). It can be shown that every simple algebraic extension of F is isomorphic to a field of this form. See the primitive element theorem.
When F is a field, the set F((X)) of formal Laurent series over F is a field.
If V is an algebraic variety over F, then the rational functions V → F form a field, the function field of V.
If S is a Riemann surface, then the meromorphic functions S → C form a field.
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.
Hyperreal numbers and superreal numbers extend the real numbers with the addition of infinitesimal and infinite numbers.

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 $2^{2^{n}}$ , the nimbers with birthday smaller than any infinite cardinal are all examples of fields.

Some first theorems

The set of non-zero elements of a field F (typically denoted by F^×) is an abelian group under multiplication. Every finite subgroup of F^× is cyclic.

The number of elements of any finite field is a prime power.

If there are positive integers n such that 0 = 1 + 1 + ... + 1 (n repeated terms), then the smallest such n must be a prime number; that is, the characteristic of a field must be either a prime number, or zero.

A commutative ring is a field if and only if it has no ideals except {0} and itself.

Assuming the axiom of choice, for every field F, there exists a field G which contains F, is unique up to isomorphism inducing the identity on F, is algebraic over F, and is algebraically closed. G is called the algebraic closure of F. However, in many circumstances in mathematics, it is not appropriate to treat G as being uniquely determined by F, since the isomorphism above is not itself unique. In these cases, one refers to such a G as an algebraic closure of F.

References

^ J J O'Connor and E F Robertson, The development of Ring Theory, September 2004.

External links

Fields at ProvenMath definition and basic properties.
"Field". PlanetMath.

[1] J J O'Connor and E F Robertson, The development of Ring Theory, September 2004.

[1]