Hello there, I'm having some trouble solving a problem. Would be really grateful if someone could lend a hand.

Prove the theorem: Given a,b,c,d belonging to Z, exist u,v belonging to Z so that (a^2+b^2)(c^2+d^2)=u^2+v^2

You're supposed to use complex numbers to solve the exercise.

I started as following: (a^2+b^2)(c^2+d^2)=U^2+v^2 <=>

<=> (a+ib)(a-ib)(c+id)(c-id)=(u+iv)(u-iv) <=>

[assuming Z1=a+ib, Z2=c+id and w=u+iv] Z1Z1Z2Z2=ww(also, assume the underline is actually an overline, meaning the conjugate)

I'm not sure of what to do next :/