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

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In summary, the conversation is about quadratic-like residues in (Z/pZ)[i], specifically if there exists a 'z' in (Z/pZ)[i] for a given non-zero element 'a' of Z/pZ such that 'a = z\overline{z}'. The responder suggests trying to mimic the proof for the fact that a prime = 1 mod 4 is the sum of two squares in Z, and also recommends looking into "formally real fields" for related material.
  • #1
zack_vt
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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 = [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
 
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  • #2


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


Many thanks!
 

1. What is a quadratic-like residue in (Z/pZ)?

A quadratic-like residue in (Z/pZ) refers to an element in the finite field (Z/pZ) that satisfies a quadratic equation of the form x^2 ≡ a (mod p), where p is a prime number and x and a are elements in (Z/pZ).

2. How do you find quadratic-like residues in (Z/pZ)?

To find quadratic-like residues in (Z/pZ), you can use the legendre symbol or the jacobi symbol. These symbols are defined as (a/p) = a^((p-1)/2) (mod p) and (a/n) = a^((n-1)/2) (mod n), respectively. If the result of these symbols is 1, then a is a quadratic-like residue in (Z/pZ).

3. What is the significance of quadratic-like residues in (Z/pZ)?

Quadratic-like residues in (Z/pZ) have various applications in number theory and cryptography. They are used in algorithms for primality testing, factorization, and encryption.

4. Can any element in (Z/pZ) be a quadratic-like residue?

No, not every element in (Z/pZ) can be a quadratic-like residue. For an element to be a quadratic-like residue, it must satisfy certain conditions, such as being relatively prime to p and having a square root in (Z/pZ).

5. How are quadratic-like residues related to quadratic residues?

Quadratic-like residues are a generalization of quadratic residues. While quadratic residues are restricted to the set of integers, quadratic-like residues can be defined in any finite field. Additionally, quadratic-like residues have similar properties and can be used in similar applications as quadratic residues.

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