Thanks for that reference, Gareth. Boy, things are a lot more complicated with factor rings of Z[i] than I would have guessed! But also very interesting. Now I wonder what's known in general about factor rings of Z[w_n] where w_n = exp(2pi*i/n) . That paper gives a lot of hints for when w_n is quadratic, i.e., its minimal polynomial is degree 2. —Dan
On May 14, 2016, at 2:55 PM, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
On 14/05/2016 19:11, Dan Asimov wrote:
So another way to phrase what Gareth is saying is that modular arithmetic is the ring of Gaussian integers modulo an ideal. Right?
Right. Just as "ordinary" modular arithmetic means working in the ring of integers modulo an ideal.
If the (an) element of Z[i] generating the ideal is a + bi, a, b in Z, then the ideal will have a^2 + b^2 elements.
Question: Is there a simple way to say which finite ring Z[i] / (a + bi) is isomorphic to? Like maybe it's always (isomorphic to) a product of rings of form Z/(n) ?
I don't know, but that is no evidence that others don't because to an excellent first approximation I don't know anything. I would guess the answer to your second question is no.
There's some analysis of this here: http://home.wlu.edu/~dresdeng/papers/factorrings.pdf which appears to give a complete answer to your question. (And indeed you don't always get a product of Z/(n).)
P.S. Is there a classification of finite commutative rings with multiplicative identity?
There's some discussion of this here: http://mathoverflow.net/questions/7133 (it seems the answer is that no classification is known).
-- g
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