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

1. ### zack_vt

6
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

2. ### morphism

2,020
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.)

3. ### zack_vt

6
Re: Question regarding quadratic-like residues in (Z/pZ).

Many thanks!