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.