Y.S.Han Finite fields 9 Properties of Fields ·a∈F,a.0=0.a=0. Proof::a=a·1=a·(1+0)=a+a.0. 0=-a+a=-a+(a+a.0.Hence,.0=0+a.0=a.0. ·Let∀a,b∈F and a,b≠0.Then a.b≠0. ·a.b=0anda≠0 imply that b=0. ·a,b∈F,-(a·b)=(-a).b=a(-b). Proof:0=0.6=(a+(-a)).b=a.6+(-a).b.Similarly,we can prove that-(a·b)=a·(-b. Cancellation law:a0 and a.b=a.c imply that b=c. Proof:Since a≠0，a1.(a:b)=a-1.(ac.Hence, (a-1·a)·b=(a-1·a)c,i.e,b=c. School of Electrical Engineering Intelligentization,Dongguan University of Technology Y. S. Han Finite fields 9 Properties of Fields • ∀a ∈ F, a · 0 = 0 · a = 0. Proof: a = a · 1 = a · (1 + 0) = a + a · 0. 0 = −a + a = −a + (a + a · 0). Hence, 0 = 0 + a · 0 = a · 0. • Let ∀a, b ∈ F and a, b ̸= 0. Then a · b ̸= 0. • a · b = 0 and a ̸= 0 imply that b = 0. • ∀a, b ∈ F, −(a · b) = (−a) · b = a · (−b). Proof: 0 = 0 · b = (a + (−a)) · b = a · b + (−a) · b. Similarly, we can prove that −(a · b) = a · (−b). • Cancellation law: a ̸= 0 and a · b = a · c imply that b = c. Proof: Since a ̸= 0, a −1 · (a · b) = a −1 · (a · c). Hence, (a −1 · a) · b = (a −1 · a) · c, i.e., b = c. School of Electrical Engineering & Intelligentization, Dongguan University of Technology