Why is the ring of Eisenstein integers interesting

Question

Also there's quite little information on its history. Can anyone please enlighten me in this ring. Particularly i had been researching about it but I feel its not quite enough to be a motivation to be researched upon.

Answer 1

It's the only (integer ring of a) quadratic extension with $6$ roots of unity. And that's the most you can have. The Gaussian integers have $4$. Everyone else has $2$. Isn't it amazing that 6th roots were packed into a mere quadratic extension? The norm just squeezes in to that little gap that makes it possible.