Factorizing Gaussian Integers

Question

It is wellknown that $n=ab\in\mathbb{Z}, a,b\in\mathbb{Z}, a,b\gt1 \implies a\le \sqrt n \lor b\le\sqrt n$.

Let $z=a+ib$ be a nonprime Gaussian integer, such that $z=(c+id)(e+if)$.

Do we have that either $|c+id|\le\sqrt{|a+ib|}$ or $|e+if|\le\sqrt{|a+ib|}$?

Answer 1

Once you take the modulus then you have converted everything back into integers so it will still hold.

Answer 2

Yes.

Let $N(z)$ be the norm of the complex number $z$, i.e. $N(a+bi) = a^2+b^2$. The norm is multiplicative in the complex numbers (one could also think of this as the square of the magnitude). If

$$z = w_1w_2$$

then

$$N(z) = N(w_1)N(w_2)$$

and because of the first statement in your question, either

$$N(w_1) \leq \sqrt{N(z)} \text{ or } N(w_2) \leq \sqrt{N(z)}$$

Taking the square root, we get that

$$|w_1| \leq \sqrt{|z|} \text{ or } |w_2| \leq \sqrt{|z|}$$

finishing the proof.