Coset representatives for $\mathcal{O}_K/(\alpha)$.

Let $\mathcal{O}_K$ be the ring of integers of a quadratic extension of $\mathbb{Q}$, and $\alpha$ some nonzero integer. I recently asked why $N(\alpha)=|\mathcal{O}_K/(\alpha)|$, and KcD seemed to imply that there was some approach that led to a set of representatives for $\mathcal{O}_K/(\alpha)$. Greg Martin suggested that $\{0,1,2,…,N(α)−1\}$ would be a complete set of representatives in the Gaussian integer case for $\alpha=a+bi,$ with $(a,b)=1$, though he didn't eludicate why.

By the Chinese remainder theorem it seems sufficient to consider $\alpha$ some prime power. How can one find a set of coset representatives mod $\alpha$?

Solution 1:

In the quadratic case there is a well-known test for showing that a module is an ideal, e.g. see section 2.3 for Lemmermeyer's notes. Then writing the ideal as a $\mathbb Z$-module in Hermite normal form makes the norm obvious - see below. For complete details in the higher degree case see the discussion on Hermite and Smith normal forms in Henri Cohen's A Course in Computational Number Theory.