What's are all the prime elements in Gaussian integers $\mathbb{Z}[i]$
So far, I know if $p$ is a rational prime, then
$(1)$ if $p\equiv 3\mod4$, then $p$ is prime in $\mathbb{Z}[i]$.
$(2)$ If $p\equiv1\mod4$ then $p=π_1 π_2$ where $π_1 $ and $π_2$ are conjugate, Then $π_1 $ and $π_2$ are primes in $\mathbb{Z}[i]$.
$(3)$ $2=(1+i)(1-i)$, then $(1+i)$and$(1-i)$ are primes in $\mathbb{Z}[i]$.
What's are all the prime elements in Gaussian integers $\mathbb{Z}[i]$? For example, $-3$ are prime in $\mathbb{Z}[i]$, but not in the above $3$ cases.
Solution 1:
Instead of prime elements one should rather talk about prime ideals. Your list then becomes
- $(p)$ is a prime ideal if $p\equiv 3\pmod 4$ is a rational prime
- $(p)=(\pi_1)(\pi_2)$ with conjugate (and distinct) prime ideals $(\pi_1)$ and $(\pi_2)$ if $p\equiv 1\pmod 4$
- $(2)=(1+i)^2$ is the square of a prime ideal.
No other ideals in $\mathbb Z[i]$ are prime. Since $\mathbb Z[i]$ is a principal ideal domain, we may call any generator of a prime ideal a prime element, and such generators are detemined only up to a unit, the units in $\mathbb Z[i]$ being $\{1,-1,i,-i\}$. So $(3)=(3i)=(-3)=(-3i)$, i.e. the primes $3, 3i, -3, -3i$ are associated primes. On the other hand $$\begin{align}&(3+4i)=(-4+3i)=(-3-4i)=(4-3i)\\\ne&(3-4i)=(4+3i)=(-3+4i)=(-4-3i),\end{align}$$ so we obtain two times four associated primes dividing $5$. Finally $(1+i)=(1-i)=(-1+i)=(-1-i)$, so we get four associated primes above $2$.