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jmogarrio
Regular poster

Post Number: 27
Posted on Wednesday, 09 May, 2012 - 06:29 pm:   

I'm being asked to describe the residue ring [inline]\mathbb{Z}[i]/(p)[/inline] for each prime [inline]p[/inline] (where I suppose [inline]p[/inline] refers to an integer prime).

I can only guess that there will be some distinction between the primes that are Gauss primes and the ones which are a product of complex conjugate Gauss primes. What the distinction is, however, and how to begin describing these rings completely eludes me.
jmogarrio
Regular poster

Post Number: 28
Posted on Wednesday, 09 May, 2012 - 06:34 pm:   

Ok, well, getting a bit hasty there... I know it's a field when [inline]p[/inline] is a Gauss prime. I don't know how to figure out the order of the field, which I suppose would be finite.

The other rings I don't understand.
Yatir Halevi
Veteran poster

Post Number: 1358
Posted on Sunday, 13 May, 2012 - 05:30 am:   

Remember that if P is a gaussian prime then its norm is a prime, p (a regular prime). I would try to prove that it is a field of p elements :-)

If you're having problems, post again.

Yatir
Billy Woods
Veteran poster

Post Number: 1060
Posted on Tuesday, 15 May, 2012 - 02:25 pm:   

Try drawing a picture? Z[i] looks like a square lattice - draw the elements of this lattice with lines between them. Write out a few small elements of the ideal (p), and then colour them in. This will look like a (possibly enlarged and rotated) copy of Z[i]. Again draw in the lines between them. I've drawn you an example of when p = 2 + 3i.

Z[i]/(2+3i)

Now quotient by this ideal - that is, roll up this picture so that all the blue points are the same. You end up with one blue point (zero) and 12 black points (non-zero elements). Multiplication and addition are exactly as they are in Z[i], except for this subsequent 'rolling up'. What ring is this?

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