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

Arithmetics in the ring \(\mathbb Z[\omega]\)

Here \(\omega\) denotes a primitive cubic root of unity. Then the norm of an element \(a+b\omega\in\mathbb{Z}[\omega]\) (\(a,b\in\mathbb{Z}\)) is \(N(a+b\omega)=a^2-ab+b^2\) and the units are \(\pm1\), \(\pm \omega\) and \(\pm(1+\omega)=\mp\omega^2\).

Theorem 8 FTA holds in the ring \(\mathbb{Z}[\omega]\).

Problem 5 If \(p\equiv1\) (mod 6) is a prime number, prove that there exist \(a,b\in\mathbb{Z}\) such that \(p=a^2-ab+b^2\).

Theorem 9 Element \(x\in\mathbb{Z}[\omega]\) is prime if and only if \(N(x)\) is prime or \(|x|\) is a prime integer of the form \(3k-1\), \(k\in\mathbb{N}\).

Maybe the most famous application of the elementary arithmetic of the ring \(\mathbb{Z}[\omega]\) is the Last Fermat Theorem for the exponent \(n=3\). This is not unexpected, having in mind that \(x^3+y^3\) factorizes over \(\mathbb{Z}[\omega]\) into linear factors: \[x^3+y^3=(x+y)(x+\omega y)(x+\omega^2y)=(x+y)(\omega x+\omega^2y) (\omega^2x+\omega y).\quad\quad\quad\quad\quad(1)\] The proof we present was first given by Gauss.

Theorem 10 The equation \[x^3+y^3=z^3\quad\quad\quad\quad\quad(\ast)\] has no nontrivial solutions in \(\mathbb{Z}[\omega]\), and consequently has none in \(\mathbb{Z}\) either.