Field

This article gives a basic definition in the following area: field theory
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Definition

A field is a set (that we'll call $k$ ) equipped with the following:

An (infix) binary operation $+$ , called the addition
An (infix) binary operation $*$ , called the multiplication
Two distinct constants $0,1\in k$ , called zero and one
A unary operation denoted by the prefix symbol $-$ (called the negative, or additive inverse)
A map ${}^{-1}:k\setminus \{0\}\to k\setminus \{0\}$

such that the following compatibility conditions hold:

$k$ forms an abelian group with binary operation $+$ , inverse map $-$ , and identity element $0$ . This is called the additive group of $k$ .
$*$ gives an associative binary operation on $k$
The following distributivity law holds:

$a*(b+c)=(a*b)+(a*c)$

$k^{*}=k\setminus 0$ forms an abelian group with binary operation $*$ , inverse map ${}^{-1}$ and identity element $1$ . This is called the multiplicative group of $k$ .

Examples

Finite fields

Further information: Finite field

It turns out that finite fields, i.e. fields with a finite number of elements, only exist with order a prime power. Furthermore, there is a unique field of said order up to isomorphism, which is usually denoted $\mathbb {F} _{q}$ or $GF(q)$ for $q$ the order, a prime power. Here are the smallest such examples:

Infinite fields

The rational numbers, real numbers and complex numbers are fields under addition and multiplication.

Concepts in field theory

See also: Field theory

If $L$ is a field, and $K$ is a subset of $L$ which is also a field, we say $K$ is a subfield and that $L$ extends $K$ . This leads to the concept of a field extension, foundationally important for field theory and Galois theory among other mathematical fields.