Factorization of Gauss integers

Question

How do I find the factorization in prime elements of $20538 - 110334i$ in $\Bbb{Z}[i]$? I have found that $20538=2×3^2×7×163$ and $110334=2×3×7×2627$ but I don't know how to use this.

Answer 1

Extract the factors $2\times 3 \times 7$ which you know. Note that one of these is not prime in your context, so there is more work to do (I assume you know what happens to integer primes in $\mathbb Z[i]$)

You are left with $489-2627i$.

Consider $(489-2627i)(489+2627i)=239,121+6,901,129=7,140,250$

Every prime factor of $489-2627i$ will be a prime factor of $7,140,250$ as will its complex conjugate. So you should be able to determine candidate factors by factorising in $\mathbb Z$ to start with. You then need to test whether it is the factor or its conjugate which is a factor of your original number.

Answer 2

$a+bi\in\mathbb{Z}[i]$ is prime if one of the following holds:

1. One of $a$ and $b$ is zero and the other is a prime in $\mathbb{Z}$ that is $3$ mod $4$.
2. $N(a+bi)=a^2+b^2$ is a prime in $\mathbb{Z}$.

You can use this to find all the prime of $\mathbb{Z}[i]$ and apply any technique that works for factoring integers except with this new list of primes to factor numbers in $\mathbb{Z}[i]$.

You can also use properties of the norm. $N(xy)=N(x)N(y)$ and $x|N(x)$ for every $x,y$. If you have $x\in\mathbb{Z}[i]$ and you factor it’s norm into primes, you can look at the numbers whose norm is those primes.