Question regarding quadratic-like residues in (Z/pZ) .

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The discussion focuses on the existence of an element 'z' in the set (Z/pZ)[i] such that a given non-zero element 'a' in Z/pZ can be expressed as 'a = z\overline{z}'. The inquiry specifically pertains to cases where 'p' is a prime number congruent to 3 mod 4. A participant clarifies that the question relates to whether 'a' can be represented as the sum of two squares in Z/pZ, which holds true regardless of the condition on 'p'. The conversation suggests exploring the properties of formally real fields for further understanding.

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Question regarding quadratic-like residues in (Z/pZ).

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 = \sqrt{-1}: (Z/pZ). I want to know if the exists a 'z' in (Z/pZ) for a given non-zero element 'a' of Z/pZ such that 'a = z\overline{z}'. If anyone could point me in a fruitful direction on this I would be most grateful.

-Z
 
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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.)
 


Many thanks!
 

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