I was studying the Gaussian integers and I proved that every composite number in $\mathbb{N}$ is not a prime in $\mathbb{Z}[i]$. This is true because this ring is an Euclidean domain, and if $n=ab$ is composite then $$\lambda(n)=\lambda(a)\lambda(b),$$ and $\lambda(a),\lambda(b)>1$ where $\lambda(\alpha)=\alpha\overline{\alpha}$. Basically I used the property that every irreducible is prime.

This same property is not valid in $\mathbb{Z}[i\sqrt5]$ because for example $2$ is irreducible but is not prime, as $2$ divides $6=(1+i\sqrt5)(1-i\sqrt5)$ but $2$ does not divide either of the factors.

How could I prove that there is some composite number of $\mathbb{N}$ which is prime in $\mathbb{Z}[i\sqrt5]$? Is it possible to list the primes of this ring?