**field**is a set \( F \) with two binary operations \( \oplus, \otimes \) such that for any \( a, b, c \in F \) we have

- \( \oplus, \otimes \) are both associative and commutative
- Identities: There exist identity elements \( 0 _ F \neq 1 _ F \in F\) for \( \oplus \) and \( \otimes \) respectively
- Additive Inverses: There exists an \( -a \in F \) such that \( a \oplus -a = 0 _ F \)
- Multiplicative Inverses: If \( a \neq 0 \) there exists an \( a ^ { -1 } \in F \) such that \( a \otimes a ^ { -1 } = 1 _ F \)
- Distributivity: \( \otimes \) distributes into \( \oplus \)