Arithmetic in Extensions of \(\mathbb Q\) (Table of contents)

Arithmetics in Gaussian Integers \(\mathbb Z[i]\)

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 \(\pm1,\pm i,\pm\pi,\pm i\pi\).

Theorem 6 The Fundamental Theorem of Arithmetic (FTA) holds in the set of Gaussian integers \(\mathbb{Z}[i]\).

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

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 \(z=ac+bd\).