Answer to Question #44365 in Abstract Algebra for Jasvinder Singh

Question

Check whether the following pairs of elements are associates:
i) 5+4i and 5

Answer 1

If a divides b and b divides a, then we say a and b are associated elements or associates. Consider a field of complex numbers. Calculate (5+4i)(5-4i)=25-16i²=25+16=41, hence (5+4i)*5(5-4i)/41=5, which means that (5+4i) divides 5.
Besides, 5*(1+4i/5)=5+4i, which means that 5 divides 5+4i. Therefore, 5+4i and 5 are associates.