Theorem 6.30: A finite integral domain is a field Example: Zm;+, is a field iff ns a prime number ◆ integral domain? 蒸◆ If GCD(a,n)=1, then there exist k and s, s.t. akins=l where k. sez ◆ns=1-ak ◆[1l=ak=[alk ◆[k]=|a1 ◆ Euclidean algorithm Theorem 6.30: A finite integral domain is a field.  Example: [Zm;+,*] is a field iff m is a prime number  Iintegral domain?  [a]-1=?  If GCD(a,n)=1,then there exist k and s, s.t. ak+ns=1, where k, sZ.  ns=1-ak.  [1]=[ak]=[a][k]  [k]= [a]-1  Euclidean algorithm