A
field, in
abstract algebra, is an algebraic system of elements in which the
operations of addition, subtraction, multiplication, and division (except division by zero) may be performed without leaving the system and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary
numbers. Fields are important objects of study in
abstract algebra since they provide the proper generalization of number domains, such as the sets of
rational numbers or
real numbers.
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:
- 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) and (b + c) * a = (b * a) + (c * a).
- Existence of an additive identity
- There exists an element 0 in F, such that for all a belonging to F, a + 0 = a and 0 + a= 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 and 1 * a = a.
- Existence of an additive inverse
- For all a belonging to F, there exists an element -a in F, such that a + (-a) = 0 and (-a) + a = 0.
- Existence of a multiplicative inverse
- For all a <> 0 belonging to F, there exists an element a^{-1} in F, such that a * a^{-1} = 1 and a^{-1} * a = 1.
The requirement 0 ≠ 1 ensures that the set which only contains a single zero is not a field. Directly from these axioms, one may show that (F, +) and (F - {0}, *) are commutative groups and that therefore the additive inverse -a and the multiplicative invers a^{-1} are uniquely determined by a. Furthermore, the inverse of a product is equal to the product of the inverses (see elementary group theory).
Examples of Fields
- The smallest field has only two elements: 0 and 1. It is sometimes denoted by F_{2} or Z_{2} and can be defined by the two tables
+ 0 1 * 0 1
0 0 1 0 0 0
1 1 0 1 0 1
- It has important uses in computer science, especially in cryptography and [coding theory]?.
- More generally: if q > 1 is a power of a prime number, then there exists (up to isomorphism) exactly one finite field with q elements. No other finite fields exist. For instance, for a prime number p, the set of integers modulo p is a finite field with p elements: this is often written as Z_{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.
- The complex numbers contain the field of algebraic numbers, the [algebraic closure]? of Q.
- The rational numbers can be extended to the fields of p-adic numbers for every prime number p.
- 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 defined as the set of quotients of polynomials with coefficients in F.
- Let E and F be two fields with E a subfield of F (i.e., a subset of F containing 0 and 1, closed under the operations + and * of F and with its own operations defined by restriction). Let x be an element of F not in E. Then E(x) is defined to be the smallest subfield of F containing E and x. For instance, Q(i) is the subfield of the complex numbers C consisting of all numbers of the form a+bi where both a and b are rational numbers.
- 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)) is a field with a subfield isomorphic to F. For instance, R[X]/(X^{2}+1) is a field (in fact, it is isomorphic to the field of complex numbers).
- The hyperreal numbers form a field containing the reals, plus infinitesimal and infinite numbers.
- The surreal numbers form a field containing the reals, 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 number form a field.
- The nimbers? form a field, again except for the fact that they are a proper class. 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.
Further properties, definitions and facts
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:
- E contains a non-zero element
- For every x and y in E, x - y is in E
- For every x and y in E with y ≠ 0, x / y is in E.
For example,
Q is a subfield of
R, which in turn is a subfield of
C.
The set of non-zero 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 an n exists, we say the characteristic is zero. Every non-zero characteristic is a prime number. For example, the rational numbers, the real numbers and the p-adic 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, p-adic 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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