Units in ring theory

Question

How to determine $U( Z[i])$

I tried this

$$(a+bi)(c+di) = 1,$$

where $a,b,c,d$ are integers. and compared real no. With real

Answer 1

Hint: note that if $a+bi$ is invertible, then the corresponding complex number must have norm $1$.

(In general, considering the norm is very often a good thing to do in this ring.)

Answer 2

You can show that if $(a+bi)(c+di)$ is an integer, then $c=a, d=-a$. So the problem reduces to solving $$(a+bi)(a-bi)=a^2+b^2=1$$

What can you say about the integers $a,b$ if the above should be true?