I was asked to show that if any Gaussian prime $p$ divides the product $abc$ of Gaussian primes, then $p$ is equal to one of those primes, or one of them multiplied by a unit.
I know that a Gaussian prime are primes with a norm that is a real prime of the form $4n+3$. It seems to me that the norm of an integer $a+bi$ is $a^2+b^2$. How can I use this information to help me solve the given problem? Will finding the norms help?
DISCLAIMER I am unsure what have been told you can assume about Gaussian integers (for example that they form a Euclidean Ring) but this is a reasonably basic proof.
A Gaussian prime is one of $\pm q,\pm qi\text { or } u+vi$ where $q$ is an integer prime of the form $4k+3$ and $u^2+v^2$ is either $2$ or an integer prime $r$ of the form $4k+1$. Therefore the norm of a Gaussian prime is of the form either $2,q^2$ or $r$.
Now suppose the Gaussian prime $p$ is a factor if the product $abc$ of three Gaussian primes. The $N(p)$ is a factor of $N(abc)=N(a)N(b)N(c)$.
If $p$ is one of $\pm q,\pm qi$ then one of $a,b,c$ must have norm $q^2$ and therefore be of the form $\pm q,\pm qi$. Then this prime is a multiple of $p$ by an element of $\{\pm1,\pm i\}$ and all these elements are units.
If $p=u+vi$ where $u^2+v^2$ is an integer prime $r$ of the form $4k+1$ (and similarly if $u^2+v^2=2$), then one of $a,b,c$ is $x+yi$ where $x^2+y^2=r$. Then $$(u-vi)p=r=(x+yi)(x-yi)$$ and so the prime $p$ is a divisor of either $x+yi$ or $x-yi$. Without loss of generality let it be a divisor of $x+yi$. Then $\frac {x+yi}{p}=l+mi$ is a Gaussian integer with norm $1$.
Therefore $l^2+m^2=1$ and so $l+mi\in \{\pm1,\pm i\}$ is a unit.