Prove Lemma 47.2.

Data from Lemma 47.2

In Z[i], the following properties of the norm function N hold for all α, β ∈ Z[i]:

N(α) ≥ 0. 

N(α) = 0 if and only if α = 0. 

N(αβ) = N(α)N(β). 

Proof If we let α = a₁ + a₂i and β = b₁ + b₂i, these results are all straightforward computations. We leave the proof of these properties as an exercise.