A field is a set S with two operations called addition \({\oplus: S\times S \rightarrow S}\) and multiplication \({\odot: S\times S \rightarrow S}\) which satisfies the following requirements:

- associativity
$$\forall_{a, b, c \in S}: (a \odot b) \odot c = a \odot ( b \odot c)$$
$$\forall_{a, b, c \in S}: (a \oplus b) \oplus c = a \oplus ( b \oplus c)$$

- commutativity
$$\forall_{ a, b \in S}: \; a \odot b = b \odot a $$
$$\forall_{ a, b \in S}: \; a \oplus b = b \oplus a $$

- identity elements
$$\exists_{ \mathrm{1} \in S} \forall_{ a \in S}: \; a \odot \mathrm{1} = \mathrm{1} \odot a = a$$
$$\exists_{ \mathrm{0} \in S} \forall_{ a \in S}: \; a \oplus \mathrm{0} = \mathrm{0} \oplus a = a$$

- inverse elements
$$\forall_{ a \neq \mathrm{0} \in S} \exists_{a^{-1} \in S}: \; a \odot a^{-1} = a^{-1} \odot a = \mathrm{1}$$
$$\forall_{ a \in S} \exists_{(-a) \in S}: \; a \oplus (-a) = (-a) \oplus a = \mathrm{0}$$

- multiplication is distributive over addition
$$\forall_{ a ,b,c \in S}: \; a \odot( b\oplus c)= ( a \odot b )\oplus (a \odot c)$$