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.
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 F2 or Z2 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 Zp = {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.
- 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]/(X2+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 and facts.
Many of the fields important in analysis are ordered fields and therefore topological spaces.
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 real numbers and the p-adic numbers have characteristic 0, while the finite field Zp 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 ring homomorphism F ->F, the "Frobenius homomorphism".
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