Fields

Tip

A $field$ is a group under both addition and multiplication

A $field$ is a set $F$ together with two binary operations $+$ and $∙$ of $F$ with the following properties
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 $a \times b = b \times a$
Additive and Multiplicative Identity: There exists two different elements $0$ and $1$ in $F$ such that
$a + 0 = a$ and $a \times 1 = a$
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