This is the second problem in Neukirch's Algebraic Number Theory. I did the proof but it feels a bit too slick and I feel I may be missing some subtlety, can someone check it over real quick?
Show that, in the ring $\mathbb{Z}[i]$, the relation $\alpha\beta =\varepsilon\gamma ^n$, for $\alpha,\beta$ relatively prime numbers and $\varepsilon$ a unit, implies $\alpha =\varepsilon '\xi ^n$ and $\beta =\varepsilon ''\eta ^n$, with $\varepsilon '$,$\varepsilon ''$ units.
So basically, because the Gaussian integers are a unique factorization domain and alpha and beta are relatively prime, I have the prime decomposition:
$\alpha = \varepsilon' p_1^{e_1}...p_r^{e_r}$
$\beta = \varepsilon'' p_s^{e_s}...p_y^{e_y}$
$\varepsilon\gamma^n = \varepsilon q_1^{nf_1}...q_k^{nf_k}$
And so $\alpha\beta = p_1^{e_1}...p_r^{e_r}p_s^{e_s}...p_y^{e_y}$. Where we have a one-to-one correspondence between the $p_i^{e_i}$ and the $q_i^{nf_i}$ and thus setting $p_i^{e_i} = q_i^{f_i}$, in accordance with this correspondence, we obtain our desired xi and eta.
Does this make sense? I never used anything specific to the Gaussian integers so if this is right then it holds for all UFDs. Thanks.