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A field, in abstract algebra, is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers.
Fields are important objects of study in algebra since they provide the proper generalization of number domains, such as the sets of rational numbers, real numbers, or complex numbers. Fields used to be called rational domains.
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. Galois theory studies the symmetry of equations by investigating the ways in which fields can be contained in each other.
Definition: 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.
Spelled out, this means that the following hold:
The requirement 0 ≠ 1 ensures that the set which only contains a single zero is not a field. Directly from the axioms, one may show that (F, +) and (F  {0}, *) are commutative groups and that therefore (see elementary group theory) the additive inverse a and the multiplicative inverse a^{1} are uniquely determined by a. Furthermore, the multiplicative inverse of a product is equal to the product of the inverses:
+ 0 1 * 0 1 0 0 1 0 0 0 1 1 0 1 0 1
A field homomorphism between two fields E and F is a function f : E > F such that f(x + y) = f(x) + f(y) and f(xy) = f(x) f(y) for all x, y in E, as well as f(1) = 1. These properties imply that f(0) = 0, f(x^{1}) = f(x)^{1} for x in E with x ≠ 0, and that f is injective. Fields, together with these homomorphisms, form a category. Two fields E and F are called isomorphic if there exists a bijective homomorphism f : E > F. The two fields are then identical for all practical purposes.
A subfield of a field F is a subset of F which is closed under the field operation + and * of F and which, with these operations, forms itself a field. Such a subfield automatically has the same additive and multiplicative identities as F, and the additive and multiplicative inverses of an element of the subfield are the same as those of the same element in F. In order to check that a subset E of F is a subfield of F, one only has to check three properties:
The set of nonzero elements of a field F is typically denoted by F^{×}; it is an abelian group under multiplication. Every finite subgroup of F^{×} is cyclic.
For every field F, there exists a (up to isomorphism) unique field G which contains F, is algebraic over F, and is algebraically closed. G is called the algebraic closure or F.
The characteristic of the field F is the smallest positive integer n such that n·1 = 0; here n·1 stands for n summands 1 + 1 + 1 + ... + 1. If no such n exists, we say the characteristic is zero. Every nonzero characteristic is a prime number. For example, the rational numbers, the real numbers and the padic numbers have characteristic 0, while the finite field Z_{p} has characteristic p.
If the characteristic of the field F is equal to the prime p, then p·x = 0 for every x in F, and (x + y)^{ p} = x^{ p} + y^{ p} for all x, y in F, a consequence of the binomial theorem. The map f(x) = x^{ p} is a field homomorphism F >F, the "Frobenius homomorphism".
Every field has a unique smallest subfield, which is called the prime subfield and is contained in every other subfield. For fields of characteristic 0, the prime subfield is isomorphic to Q (the rationals). Fields of characteristic 0 are therefore always infinite. For fields of prime characteristic p, the prime subfield is isomorphic to Z_{p}. Fields of prime characteristic can be either infinite or finite (see Finite field).
All the fields of importance in analysis (real numbers, complex numbers, padic numbers, nonstandard reals) carry a valuation[?] or an order, which turns them into topological spaces; addition, subtraction, multiplication and division are then continuous operations. All these fields have characteristic zero.
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