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.

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