determine the set of prime ideals of the Dedekind Domain

Determine the set of prime ideals of the Dedekind Domain $\mathbb{Z}[i] = \{a+bi: a,b \in \mathbb{Z}\}$ and factor the ideal $(7-i)$ into prime ideals.

I know in Dedekind Domains only proper ideals can be factored into prime ideals, so I check whether $(7-i)$ is a proper ideal.

Assume it's not proper, then it would contain $1$, which implies $\exists$ $a,b$ in $\mathbb{Z}$ such that $(a+bi)(7-1) = 1$ $\iff$ $7a+b + (7b-a) = 1$ $\iff$ $7a+b=1$ and $7b-a = 0$ which has no solutions in $\mathbb{Z}$. Hence the ideal is proper and I found a factorisation $(7-i) = (3-4i)(1+i)$.

How can I check that those two are prime ideals? And how can I determine the set of all prime ideals of $\mathbb{Z}[i]$?

Thanks for any help.

• Do you know anything about the norms of elements in this context? – Mark Bennet Sep 4 '18 at 11:33

It is known that $\mathbb{Z}[i]$ is a Euclidean domain, so all ideals are principle. This can be done using the norm function, which takes $N(a+bi) = a^2+b^2$. This is multiplicative function too, you can check this. To check if your given ideal is principle, it suffices to check that its norm is a prime number (why?). We have that $N(1+i) = 1^2+1^2 = 2$ is prime so that $1+i$ is prime. However, $N(3-4i) = 3^2+4^2 = 25$. Try factoring it using the Euclidean algorithm, I can help if you get stuck. This should also help you with all primes. I can tell you that $p$ remains prime if and only if it is 3 mod 4, try to prove this.