Y.S.Han Finite fields 7 Fields Let F be a set of elements on which two binary operations, called addition“+”and multiplication“.”,are defined.The set F together with the two binary operations and.is a field if the following conditions are satisfied: 1.(F,+)is a commutative group.The identity element with respect to addition is called the zero element or the additive identity of F and is denoted by 0. 2.(F-10),)is a commutative group.The identity element with respect to multiplication is called the unit element or the multiplicative identity of F and is denoted by 1. 3.Multiplication is distributive over addition;that is,for any three elements a,b and c in F, a·(b+c）=a.b+a·c. The order of a field is the number of elements of the field. School of Electrical Engineering Intelligentization,Dongguan University of Technology Y. S. Han Finite fields 7 Fields • Let F be a set of elements on which two binary operations, called addition “+” and multiplication “·”, are defined. The set F together with the two binary operations + and · is a field if the following conditions are satisfied: 1. (F, +) is a commutative group. The identity element with respect to addition is called the zero element or the additive identity of F and is denoted by 0. 2. (F − {0}, ·) is a commutative group. The identity element with respect to multiplication is called the unit element or the multiplicative identity of F and is denoted by 1. 3. Multiplication is distributive over addition; that is, for any three elements a, b and c in F, a · (b + c) = a · b + a · c. • The order of a field is the number of elements of the field. School of Electrical Engineering & Intelligentization, Dongguan University of Technology