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  • Definition: A field F is a set together with two laws of composition called addition (+) and multiplication (x) which satisfy these axioms: i) Addition makes F into an abelian group (F,+); it's identity element is denoted by 0. ii) Multiplication is commutative, and it makes the set of nonzero elements of F into an abelian (F,x); its identity element is denoted by 1. iii) Distributive law: For all a, b, and c in F, a(b+c) = ab + ac.

    Source: Algebra, second edition by Michael Artin

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