체 (수학)

실수와 동일한 방식으로 더하기와 곱하기가 정의되는 집합.

정의

A field is a set $F$ together with two binary operations on $F$ called addition and muliplication. A binary operation on $F$ is a mapping $F \times F \to F$ .

The addition of $a$ and $b$ in $F$ is called the sum of $a$ and $b$ , and is denoted as $a + b$ .
The multiplication of $a$ and $b$ in $F$ is called the product of $a$ and $b$ , and is demoted as $a \times b$ .

The following properties are required to be satisfied:

Associativity of addition and multiplication: $a + (b + c) = (a + b) + c$ , and $a \times (b \times c) = (a \times b) \times c$ .
Commutativity of addition and multiplication: $a + b = b + a$ , and $a \times b = b \times a$
Distributivity of multiplication over addition: $a \times (b + c) = (a \times b) + (a \times c)$
Additive identity and multiplicative identity: there exist two distinct elements 0 and 1 in $F$ such that $a + 0 = a$ and $a \times 1 = a$ .
Additive inverse: for every $a$ in $F$ , there exists an element in $F$ , denoted by $- a$ , called the additive inverse of $a$ , such that $a - a = 0$
Multiplicative inverse: for every $a \neq 0$ in $F$ , there exists an element in $F$ , denoted by $a^{- 1}$ or $1 / a$ , called the multiplicative inverse of $a$ , such that $a \cdot a^{- 1} = 1$ .