Classify the prime elements of the Gaussian integers \(\ZZ[i]\), up to multiplication by units.
Solution
The units of \(\ZZ[i]\) are \[ \{\pm1,\pm i\}. \] Up to these units, the Gaussian primes are exactly the following.
1. The ramified prime above \(2\):\[ 1+i, \]with\[ 2=-i(1+i)^2. \]
2. Rational primes \(p\equiv3\pmod4\).
Such a prime remains irreducible in \(\ZZ[i]\). If it factored nontrivially, taking norms would write \[ p^2=N(\alpha)N(\beta) \] with both norms greater than \(1\), forcing one factor to have norm \(p\). But \[ a^2+b^2=p \] has no solution when \(p\equiv3\pmod4\).
3. Gaussian integers \(a+bi\) with both coordinates nonzero and\[ a^2+b^2=p \]a rational prime.
Their norm is the rational prime \(p\). Since the norm is multiplicative, any factorization would force one factor to have norm \(1\), hence be a unit. Thus \(a+bi\) is prime.
For rational primes \(p\equiv1\pmod4\), Fermat’s two-squares theorem gives \[ p=a^2+b^2=(a+bi)(a-bi), \] so \(p\) itself is not Gaussian prime; the factors \(a\pm bi\) are.
Equivalently, a nonunit \(a+bi\) is Gaussian prime iff either - one of \(a,b\) is zero and the absolute value of the other is a rational prime congruent to \(3\pmod4\), or - both are nonzero and \(a^2+b^2\) is a rational prime.