Y.S.Han Finite fields 5 Proof:Since p is a prime,gcd(i,p)=1 for all 0<i<p.By Euclid's theorem,a,bZ such that a.i+bp=1.Then a:i=-b·p+l.If0<a<p,then a▣i=i回a=1.Assume that a >p.Then a=g.p+r,where r<p.Since gcd(a,p)=1, r≠0.Hence,r.i=-(b+q·i)p+1,i.e.,r回i=i回r=1. School of Electrical Engineering Intelligentization,Dongguan University of Technology Y. S. Han Finite fields 5 Proof: Since p is a prime, gcd(i, p) = 1 for all 0 < i < p. By Euclid’s theorem, ∃a, b ∈ Z such that a · i + b · p = 1. Then a · i = −b · p + 1. If 0 < a < p, then a ⊡ i = i ⊡ a = 1. Assume that a ≥ p. Then a = q · p + r, where r < p. Since gcd(a, p) = 1, r ̸= 0. Hence, r · i = −(b + q · i)p + 1, i.e., r ⊡ i = i ⊡ r = 1. School of Electrical Engineering & Intelligentization, Dongguan University of Technology