Trivial Factor of Gaussian Primes

Question

Hello anyone know how to prove that α=2+i is Gaussian primes? Here my step, so I let α=βγ then N(α)=N(β)N(γ), we get the norm of α which is N(α)=5 primes in Z. We get N(β)=1,N(γ)=5 or N(β)=5,N(γ)=1. Without loss of generality, we choose N(β)=1, it means β is unit and γ is the product of unit and α. So α is Gaussian primes. But my lecturer says, that my proof is not correct perfectly, I need to show what’s the form of γ, according to the definition let α is non zero element of Z[i], α is Gaussian primes if α is not unit and has trivial factor which is ±1,±i,±α, dan ±iα. I need to show the trivial factor, and it has to be like the definition what I say. Please help me, thank you.

Answer 1

I'm going to interpret this question as asking why elements with norm $1$ are units. Let $\gamma=a+ib$ and suppose $a^2+b^2=N(\gamma)=1.$ Then $0\leq a^2\leq 1$ and $0\leq b^2\leq 1.$ Either $a=\pm 1$ and $b=0$ or $a=0$ and $b=\pm 1$. Then $\gamma=1,-1,i$ or $-i$, all units.