MSE
We can view $F_7$ as the quotient ring $𝐙[ω]/(3+ω)𝐙[ω]$.
$(3+ω)$ is a maximal ideal of $𝐙[ω]$, so $𝐙[ω]/(3+ω)𝐙[ω]$ is a field.
$N(3+ω)=7$, so $𝐙[ω]/(3+ω)𝐙[ω]$ has order 7.
When we do that the non-zero elements of $F_7$ are represented by the cosets of the sixth roots of unity, i.e. the cosets $(-\omega)^j+(3+\omega)\mathbf{Z}[\omega]$. This is similar in spirit to the discussion in the linked question.
Things become more uniform, when instead of using the complex numbers we use the $p$-adic integers $\mathbf{Z}_p$ (not to be confused with the residue class ring $\mathbf{Z}/p\mathbf{Z}$. If $q=p^n$, and $\zeta$ is a root of unity of order $q-1$, then we have an isomorphism
$$
F_q=\mathbf{Z}_p[\zeta]/p\mathbf{Z}_p[\zeta]
$$
between the finite field $F_q$ and the quotient ring of an extension ring of the $p$-adic integers. That's a rather different animal, and this last paragraph may be meaningless to you unless you have the right background. The $p$-adic integeres themselves cannot be readily visualized geometrically, because the metric there is very weird.
Every finite field is a residue field of a number field