Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $\alpha$?

share|improve this question
    
Greg did not say those positive integers are a set of representatives in general for cosets of ${\mathbf Z}[i]$. Only when $\alpha$ has relatively prime real and imaginary parts do those integers represent ${\mathbf Z}[i]/(\alpha)$. –  KCd Apr 18 '12 at 17:37
    
Thanks, I've fixed it. –  Eric Gregor Apr 18 '12 at 17:56
add comment

1 Answer

up vote 2 down vote accepted

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.

enter image description here

share|improve this answer
    
Excellent, thank you. –  Eric Gregor Apr 18 '12 at 22:57
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.