$A$ is the ring of integers generated by the cube roots of unity. How will we prove that it is an Euclidean ring?
We have following results and theorem:
1) Any integral quadratic extension $A$ of $\Bbb Z$ is obtained as the ring of algebraic integers in $\mathbb Q(\sqrt d)$, [where d is a square-free integer different from 1]. The elements of $A$ are of the form $a + \sqrt d$ here $(a, b\in Z)$ if $d \equiv 2$ or $3 \pmod 4$ , and $a + b(1 + \sqrt d) / 2$ if $d \equiv 1\pmod 4$
2) The ring of integers in $\Bbb Q(\sqrt d)$ is Euclidean for $d=2,3,5,-1,-2,-3, -7, -11.$