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

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:

• Field:F2
• Field:F3
• Field:F4
• Field:F5
• Field:F7
• Field:F8
• Field:F9
• Field:F11

Infinite fields

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