体の定義は講義ノートを写したけど、英語のwikiでも同じになってる
https://en.wikipedia.org/wiki/Field_(mathematics)
Associativity of addition and multiplication: a + (b + c) = (a + b) + c, and a · (b · c) = (a · b) · c.
Commutativity of addition and multiplication: a + b = b + a, and a · b = b · a.
Additive and multiplicative identity: there exist two different elements 0 and 1 in F such that a + 0 = a and a · 1 = a.
Additive inverses: for every a in F, there exists an element in F, denoted −a, called the additive inverse of a, such that a + (−a) = 0.
Multiplicative inverses: for every a ≠ 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 · a−1 = 1.
Distributivity of multiplication over addition: a · (b + c) = (a · b) + (a · c).