2)The set is closed under multiplication,and the set faF,a0)forms an Abelian group(whose identity is called1)under the multiplication() 3)Distributive law:For all a,b,cEF,(a+b)*c=(a*c)+(b*c). 我们经常用0'表示加法运算下的单位元，‘a'表示a的加法逆。经常用1'表示乘法 运算下的单位元，a表示a的乘法逆。这样，减法a-b means sa(-b,除法a/b means b'a field thelementsif xists iscalled inte fed Galosfeld,and is denoted by GF(q). For example,GF(2)-the smallest field Let F be a field.A subset of F is called a subfield if it is a field under the inherited addition and multiplication.原来的域F称为an extension field of the subfield ■In any field,ifab=ac and a≠o,then b-c. For any positive integer m,there exists a Galois field of 2"elements,denoted by GF(2) The construction of GF(2")is very much the same as the construction of the complex-number field from the real-number field. We begin with a primitive polynomial p(x)of degree m with coefficients from the binary field GF(2). -Let a be the root of p(x);i.e.,p(a)=0.Then the field elements can be represented by (,a),where g=2 0=a,1=a°，a,a2,a-2, 1=(since ais a root of p(x)and p(x)+1,amust be a root of+1) For example:Construct GF(4)from GF(2)using p(x)=x+x+1. Polynomial notation Binary notation Integer Exponential 0 00 0 0 01 1 10 2 x+1 11 a 3 a3-1=a0 1414 2) The set is closed under multiplication, and the set {a∈F, a≠0} forms an Abelian group (whose identity is called ‘1’) under the multiplication (*); 3) Distributive law: For all a, b, c∈F, (a+b)*c=(a*c)+(b*c). 我们经常用‘0’表示加法运算下的单位元，‘-a’表示 a 的加法逆。经常用‘1’表示乘法 运算下的单位元，‘a-1’表示 a 的乘法逆。这样，减法 a-b means a+(-b), 除法 a/b means b-1a. „ A field with q elements, if it exists, is called a finite field, or a Galois field, and is denoted by GF(q). For example, (2) GF ——the smallest field „ Let F be a field. A subset of F is called a subfield if it is a field under the inherited addition and multiplication. 原来的域F称为 an extension field of the subfield. „ In any field, if ab=ac and a≠0, then b=c. „ For any positive integer m≥1,there exists a Galois field of 2m elements, denoted by (2 ) m GF . „ The construction of ) (2m GF is very much the same as the construction of the complex-number field from the real-number field. We begin with a primitive polynomial ( ) p x of degree m with coefficients from the binary field (2) GF . - Let α be the root of ( ) p x ; i.e., ( ) p α =0. Then the field elements can be represented by 2 2 {0,1, , ,., } q αα α − , where q=2m. 0 = α−∞ ,1 = 0 α , 1 α , 2 α , ., q 2 α − , 1= q−1 α (since α is a root of ( ) p x and 2 1 ( )| 1 m px x − + ,α must be a root of 2 1 1 m x − + ) - For example: Construct (4) GF from (2) GF using 2 p() 1 xxx = + + . Polynomial notation Binary notation Integer Exponential 0 00 0 0 1 01 1 1 x 10 2 α x+1 11 3 2 3 0 1 α α = =α