How do you find the quadratic residues mod p?
We only need to solve, when a number (b) has a square root modulo p, to solve quadratic equations modulo p. Given a number a, s.t., gcd(a, p) = 1; a is called a quadratic residue if x2 = a mod p has a solution otherwise it is called a quadratic non-residue.
What is a quadratic residue modulo p?
Modulo a prime p, a quadratic residue a has 1 + (a|p) roots (i.e. zero if a N p, one if a ≡ 0 (mod p), or two if a R p and gcd(a,p) = 1.)
Why are quadratic residues important?
Quadratic reciprocity is important because it provides a bridge between two apparently distinct branches of mathematics, namely the theory of Galois representations and the theory of automorphic forms.
How do you find a quadratic residue?
In other words, we have proved Euler’s Criterion, which states is a quadratic residue if and only if a ( p − 1 ) / 2 = 1 , and is a quadratic nonresidue if and only if a ( p − 1 ) / 2 = − 1 . Example: We have is a quadratic residue in if and only if p = 1 ( mod 4 ) .
Do quadratic residues form a group?
If a and b are two quadratic residues of the prime p, then it is easily checked that ab is also a quadratic residue modulo p; if c is a quadratic residue modulo p, and cd≡1(modp), then since 1 is a quadratic residue of p, d is a quadratic residue of p; so the set of all quadratic residues form a group, denoted by R.
How do you check for quadratic residue?
Is 3 a quadratic residue?
Thus, we conclude that 3 is a quadratic residue modulo p precisely when p = 2, or when p ≡ 1 or 11 (mod 12). We want to compute (6 p ) = (2 p ) · (3 p ) , for p = 2,3. (mod 12), and (3 p ) = −1 when p ≡ 5 or 7 (mod 12). 12) and p ≡ 1,7 (mod 8).
How do you prove quadratic reciprocity?
Our first proof of quadratic reciprocity is due to Eisenstein and relies on a lemma on couting lattice points. The second proof we give is also due to Eisenstein, using the nth roots of unity. After that, we will provide a recent proof given by Rousseau in 1991 [Rou91] which is quite simple and uses group theory.
When 2 is a quadratic residue?
2(p-1)/2 ≡ (−1)2k+2 ≡ 1 (mod p), so Euler’s Criterion tells us that 2 is a quadratic residue. This proves that 2 is a quadratic residue for any prime p that is congruent to 7 modulo 8.