Given a member of the gaussian integers find a all other gaussian integers with the same norm

Question

Let $\alpha$ be a non-zero non-unit member of $Z[i]$, using the fact that $Z[i]$ is a UFD, describe how to determine all the $\beta \in Z[i]$ with $N(\alpha) = N(\beta)$

Answer 1

We’ll use repeatedly the factoriality of $ \mathbf Z [i]$, as well as the decomposition of primes in this ring. Let $\alpha, \beta \in \mathbf Z [i]$ , such that $N(\alpha) = N(\beta)$. Exclude the trivial case where $\alpha = \beta $ up to units. Cancelling out common factors (which does not change the equality of norms), we may assume that $\alpha, \beta$ are coprime. The elements of norm 1 in $ \mathbf Q (i)$ , such as $\alpha. {\beta}^{-1}$, are of the form $\bar z . z^{-1}$, where $\bar z$ is the conjugate of $z$ : this is Hilbert’s theorem 90, easy to show directly in our case, and we can obviously take $z \in \mathbf Z [i]$. Let us simplify the fraction $\bar z . z^{-1}$. Denote systematically by $\pi$ a prime of $\mathbf Z [i]$ and $p$ the rational prime under $\pi$. If $p$ is inert or ramified, the powers of $\pi$ can be visibly cancelled out. Only remain the powers of a prime $\pi$ s.t. the underlying $p$ is split (i.e. $p \equiv 1 $ mod 4, but this will not be used). Thus we get $\alpha. z = \beta . \bar z$ in $ \mathbf Z [i]$, with $\alpha, \beta$ coprime and $z, \bar z$ coprime. Factoriality implies that, up to units, $\alpha = \bar z$ and $ \beta = z$. Recall that the units in $\mathbf Z [i]$ are the 4-th roots of 1. Summarizing : for two coprimes Gaussian integers, $N(\alpha) = N(\beta)$ iff $\alpha$ is equal or conjugate to $\beta$ up to 4-th roots of 1 .