Show that if alpha and beta are Gaussian integers and alpha

Show that if alpha and beta are Gaussian integers and alpha divides beta then N(alpha) divides N(beta)

Solution

= a+bi and = c+di

N() = a² +b²

N() = c² +d²

Given that   divides .

That means,

  / = k (constant)

We have to prove that  N() divides N().

i.e. N() / N() = k or some constant

N() / N() = c² +d² /  a² + b²

   = (c+id) (c-id) / (a+ib) (a-ib)

=  ²/²

= ( /)²

= (k)²

   = constant

N() / N() = constant

Therefore,

N() divides N()