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-   -   Question regarding quadratic-like residues in (Z/pZ)[i]. (http://www.physicsforums.com/showthread.php?t=567291)

zack_vt Jan13-12 12:08 PM

Question regarding quadratic-like residues in (Z/pZ)[i].
 
Hi all.

I'm working in the set that is formed by extending the integers mod p (p is prime and equal to 3 mod 4) by including i = [itex]\sqrt{-1}[/itex]: (Z/pZ)[i]. I want to know if the exists a 'z' in (Z/pZ)[i] for a given non-zero element 'a' of Z/pZ such that 'a = z[itex]\overline{z}[/itex]'. If anyone could point me in a fruitful direction on this I would be most grateful.

-Z

morphism Jan13-12 05:25 PM

Re: Question regarding quadratic-like residues in (Z/pZ)[i].
 
You're basically asking if a is the sum of two squares in Z/pZ. This is true even if p != 3 mod 4. Try to mimic the proof of the fact that a prime = 1 mod 4 is the sum of two squares in Z.

For related material, you can try reading up on "formally real fields". (Z/pZ is a nonexample.)

zack_vt Jan13-12 05:36 PM

Re: Question regarding quadratic-like residues in (Z/pZ)[i].
 
Many thanks!


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