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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?

user26857
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    recall that prime elements in any integral domain are always irreducible, so if you can show that an element is reducible then necessarily you will have that it is not prime – Atticus Stonestrom Nov 03 '20 at 16:54
  • then exploit the fact that prime and irreducible elements are the same in $\mathbb{Z}$ – Atticus Stonestrom Nov 03 '20 at 16:55
  • (hence, if an element is not prime in $\mathbb{Z}$, it will be reducible in $\mathbb{Z}$, and so... what can you conclude about it in any ring containing $\mathbb{Z}$?) – Atticus Stonestrom Nov 03 '20 at 16:56
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    @AtticusStonestrom and so it is reducible in any ring containing $\mathbb{Z}$ and so it can not be prime. Thanks!! :) –  Nov 03 '20 at 17:02

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Because $\Bbb{N}\subset\Bbb{Z}[i\sqrt{5}]$, any factorization of $n$ in $\Bbb{N}$ is also a factorization of $n$ in $\Bbb{Z}[i\sqrt{5}]$.

This relies on the fact that the ring extension $\Bbb{Z}\subset\Bbb{Z}[i\sqrt{5}]$ does not introduce any new units, as noted in the comments.

Servaes
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  • Thanks!! I have edited my question. Can I list the primes of this ring, or at least describe them? –  Nov 03 '20 at 16:51
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    You have to also note that none of the factors in $\mathbb N$ will become units in $\mathbb Z[i\sqrt{5}]$. – Wojowu Nov 03 '20 at 17:29
  • @Wojuwu You are right, I have made this explicit. – Servaes Nov 03 '20 at 17:33