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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