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February 11th, 2014, 12:26 AM  #1 
Newbie Joined: Feb 2014 Posts: 1 Thanks: 0  ring of integers problem
hi, i tried to solve the attachment problem. i tried to calculate the norm of u and its get complicate. thank you for the help 
February 19th, 2014, 04:04 AM  #2 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: ring of integers problem
I don't see how the "norm" of u has anything to do with this. You are told that u is a unit which means it is invertible. What is the multiplicative inverse of u? What must be true of a and b for that to exist?

February 22nd, 2014, 11:15 AM  #3 
Senior Member Joined: Mar 2012 Posts: 294 Thanks: 88  Re: ring of integers problem
I'm not sure where YOU are going with this. As is easy to verify, the multiplicative inverse of u is: So clearly, we require that: both be integers. But arguing from the norm: since this norm is multiplicative, it is clear that N(u) is a unit in the ring of integers, that is to say N(u) = 1 or 1. This is a much stronger condition on the coefficients than them merely being integers, so I fail to see how your observation is pertinent. Arguing mod 4, it turns out that N(u) = 1 if exactly one of a or b is even, and d = 3 (mod 4). It also turns out if N(u) = 1, then both a and b are odd, and d = 2 (mod 4). Unfortunately, I haven't been able to make further progress on this. 

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integers, problem, ring 
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