Field

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Definition

A field is a set (that we'll call k) equipped with the following:

  • An (infix) binary operation +, called the addition
  • An (infix) binary operation *, called the multiplication
  • Two distinct constants 0, 1 \in k, called zero and one
  • A unary operation denoted by the prefix symbol - (called the negative, or additive inverse)
  • A map {}^{-1}: k \setminus \{ 0 \} \to k \setminus \{ 0 \}

such that the following compatibility conditions hold:

  • k forms an abelian group with binary operation +, inverse map -, and identity element 0. This is called the additive group of k.
  • * gives an associative binary operation on k
  • The following distributivity law holds:

a * (b + c) = (a * b) + (a * c)