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Definition (Field)

A field is a set $F$ together with two binary operations $+$ and $\cdot$ that satisfies the following properties:

  1. $(F,+)$ is an Abelian group. Let $F^*$ be the set $F$ with the additive identity removed.
  2. $(F^*,\cdot)$ is an Abelian group.
  3. The distributive laws hold, namely we have $a(b+c)=ab+ac$ and $(b+c)a=ba+ca$.

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