Determine all units in the ring Zi of Gaussian Integers expl

Determine all units in the ring Z[i] of Gaussian Integers. (explain who to get 1,-1 and i, -i)

Solution

Let (a + bi) be a unit in Z[i] (for some a, b in Z).

Then, (a + bi)(c + di) = 1 for some c, d in Z.

Taking conjugates yields
(a - bi)(c - di) = 1.

Multiplying the last two equations together yields
(a^2 + b^2)(c^2 + d^2) = 1.

Since this is an equation in (positive) integers, we conclude that
a^2 + b^2 = 1 ==> (a, b) = (±1, 0) or (0, ±1).

Hence, a + bi = ±1 + 0i or 0 ± i.
(So, U(Z[i]) = {±1, ± i}.)