Arithmetic in Extensions of \(\mathbb Q\) (Table of contents)
# Arithmetics in Gaussian Integers \(\mathbb Z[i]\)

**Theorem 6**
The Fundamental Theorem of Arithmetic (FTA)
holds in the set of Gaussian integers \(\mathbb{Z}[i]\).
**Theorem 7**
An element \(x\in\mathbb{Z}[i]\) is prime if
and only if \(N(x)\) is a prime or \(|x|\) is a prime integer of the
form \(4k-1\), \(k\in\mathbb{N}\).
** Problem 3**
Solve the equation \(x^5-1=y^2\) in integers.
**Problem 4**
Suppose that \(x,y\) and \(z\) are natural
numbers satisfying \(xy=z^2+1\). Prove that there exist integers
\(a,b,c,d\) such that \(x=a^2+b^2\), \(y=c^2+d^2\) and \(z=ac+bd\).

We have already seen that the norm of element \(a+bi\in\mathbb{Z}[i]\) (\(a,b\in\mathbb{Z}\)) is \(N(a+bi)=a^2+b^2\) and the units are \(\pm1\) and \(\pm i\). Therefore, all divisors of a prime element \(\pi\in\mathbb{Z}[i]\) are \(\pm1,\pm i,\pm\pi,\pm i\pi\).

The following proposition describes all prime elements in the set of Gaussian integers.