I am considering the Euler totient function for Gaussian integers. In reference to this question, I would wish to use the fact that $\phi(p^{k})=N(p)^{k−1}\phi(p)$ if $p$ is prime, but have not succeeded in finding a proof anywhere.
Alternatively, a more specific formula is given here by $\phi(p^{k})=N(z)^k - N(z)^{k-1}$ if $N(z) \equiv_{4} 1$, but I also do not know how to rigorously show this.