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

  1. 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 = [itex]\sqrt{-1}[/itex]: (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[itex]\overline{z}[/itex]'. If anyone could point me in a fruitful direction on this I would be most grateful.

    -Z
     
  2. jcsd
  3. morphism

    morphism 2,020
    Science Advisor
    Homework Helper

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

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

    Many thanks!
     
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