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 …

A group is a set G with an operation \(\odot\) which satisfies four requirements:

- closure
$$\forall_{a, b \in G}: \; a \odot b \in G$$

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

- identity element …