Field theory

A field (F. +, *) is a set F with two operations + and * (usually called addition and multiplication), such that the following axioms hold:

  • (F, +) is an abelian group (identity element 0)
  • (F\{0}, *) is an abelian group (identity element 1)
  • Distributivity of multiplication over addition: For all a, b and c in F, the following equality holds:
    a · (b + c) = (a · b) + (a · c)

Another definition:
A field is a nonzero commutative ring that contains a multiplicative inverse for every nonzero element.