# Geometric understanding of principal/non-principal ideals

A number field $K$ with the $r$ embeddings into $\mathbb R$ and $2s$ pairs of conjugate embeddings into $\mathbb C$ can put into ring homomorphism with the product of rings $\mathbb R^r \times \mathbb C^s$, ideals of the ring of integers of $K$ are mapped to lattices in this space.

Are there any geometric criteria to tell the difference between a principal ideal and non-principal ones? Can we somehow use the geometry of these lattices to find the non-principal ideals? To find all nonprime atoms?

I got this question from the graphics here.

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