Subfield

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 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. See Field theory (mathematics) for more.

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).

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 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:

(a*b)^-1 = a^-1 * b^-1

provided both a and b are non-zero. 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.

Examples of fields

• The rational numbers Q = { a/b | a, b in Z, b ≠ 0 } where Z is the set of integers.

• The real numbers R .

• The complex numbers C.

• 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

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 real numbers contain several interesting fields: the real algebraic numbers, the computable numbers, and the definable numbers.

• 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.

• 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.

• 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.

• 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).

• 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 (using U) is a field.

• 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.

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 characteristic of any field is zero or a prime number. (The characteristic is defined as the smallest positive integer n such that n·1 = 0, or zero if no such n exists; here n·1 stands for n summands 1 + 1 + 1 + ... + 1.)

• The number of elements in finite fields is a prime power.

• As a ring, a field has no ideals except {0} and itself.

• 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.

Constructing new fields from given ones

1. If a subset E of a field (F,+,*) together with the operations *,+ restricted to E is itself a field, then it is called a subfield of F. Such a subfield has the same 0 and 1 as F.
2. The polynomial field F(x) is the field of fractions of polynomials in x with coefficients in F.
3. An algebraic extension of a field F is the smallest field containing F and a root of an irreducible polynomial p(x) in F[x]. Alternatively, it is identical to the factor ring F[x]/<p(x)>, where <p(x)> is the ideal generated by p(x).

See also

• See field theory (mathematics) for some history and other information.
• See Glossary of field theory for more definitions in field theory.
• Fields - wikibook link.