
Ask HN: Complex modulo is a division ring? Can you prove it? - AnimalMuppet
The integers mod n form a mathematical ring.  If n is prime, they form a division ring (a ring in which every number has a multiplicative inverse).<p>The complex integers modulo nr (real) and ni (imaginary) also form a ring.  They appear to form a division ring iff 1) nr = ni, and 2) nr is a Mersenne prime.<p>Does anyone know how to prove that this is in fact a division ring (other than exhaustively)?  Can anyone ELI5 (or even ELI20) why it has to be modulo a Mersenne prime?
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yesenadam
(Not an expert!) Am a bit confused what you mean "modulo nr and ni" \- modulo
Gaussian integers? Then I think "they form a division ring" means the same as
"The residue class ring modulo a Gaussian integer z is a field"? as in

"The residue class ring modulo a Gaussian integer z is a field if and only if
z is a Gaussian prime."

[https://en.wikipedia.org/wiki/Gaussian_integer#Residue_class...](https://en.wikipedia.org/wiki/Gaussian_integer#Residue_class_fields)

There's a nice picture of the situation, just above that section.

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AnimalMuppet
nr and ni meant that the real part and the imaginary part could be modulo
different numbers. For example, the real part could be modulo 7, and the
imaginary part modulo 3. If those two numbers are not the same, it turns out
not to be a division ring.

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boofgod
Can this be constructed as a quotient ring of the complex numbers by some
ideal?

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AnimalMuppet
I don't really understand ideals, so... I don't know. I _think_ so, but I
think that a quotient ring isn't quite the same as a division ring.

For a division ring, there has to be a multiplicative identity, and for every
non-zero element of the ring, there has to be a unique reciprocal.

