Describe $\mathbb{Z}[\omega] / (2)$

Question

$$\Bbb Z[ω] = \{\;a + bω: a, b\in\Bbb Z\;\}\;,\;\; ω = e^{2πi / 3} = -\frac12 +\frac{\sqrt3}2i$$

Describe $\;\Bbb Z[ω]/(2)\;$ where $\;(2)\;$ is an ideal. I already described it as a ring, and am now looking to describe it as:

x̅₁, ..., x̅_r for some specific x₁, ..., x_r where x̅_i are in the ideal.

Answer 1

Notice that $2$ and $2\omega$ are both in the ideal $(2)$. This means you can reduce any element $a+b\omega$ to one of these equivalence classes: $$\{0, 1, \omega, 1+\omega\}.$$ Now you can describe the ring structure by writing out a table for addition and a table for multiplication (keeping in mind that $1+1 = 0$ and $\omega^2 = \omega + 1$).

There aren't so many possibilities though; for instance, the multiplicative group of nonzero elements can only be a cyclic group of order three.

Answer 2

Not sure this is what you expect, but your question isn't very clear.

As $\mathbf Z[\omega]\simeq \mathbf Z[x]/(x^2+x+1)$ we have: $$\mathbf Z[\omega]/(2)\simeq \mathbf Z/2\mathbf Z[x]/(x^2+x+1)$$ The polynomial $x^2+x+1$ is irreducible over $\mathbf Z/2\mathbf Z=\mathbf F_2$, hence this quotient is a quadratic extension of $\mathbf F_2$ (there's only, up to isomorphism), i.e. it is isomorphic to $\mathbf F_4$.