# Legendre symbol #

This file contains results about Legendre symbols.

We define the Legendre symbol $\Bigl(\frac{a}{p}\Bigr)$ as legendreSym p a. Note the order of arguments! The advantage of this form is that then legendreSym p is a multiplicative map.

The Legendre symbol is used to define the Jacobi symbol, jacobiSym a b, for integers a and (odd) natural numbers b, which extends the Legendre symbol.

## Main results #

We also prove the supplementary laws that give conditions for when -1 is a square modulo a prime p: legendreSym.at_neg_one and ZMod.exists_sq_eq_neg_one_iff for -1.

See NumberTheory.LegendreSymbol.QuadraticReciprocity for the conditions when 2 and -2 are squares: legendreSym.at_two and ZMod.exists_sq_eq_two_iff for 2, legendreSym.at_neg_two and ZMod.exists_sq_eq_neg_two_iff for -2.

## Tags #

theorem ZMod.euler_criterion_units (p : ) [Fact ()] (x : (ZMod p)ˣ) :
(∃ (y : (ZMod p)ˣ), y ^ 2 = x) x ^ (p / 2) = 1

Euler's Criterion: A unit x of ZMod p is a square if and only if x ^ (p / 2) = 1.

theorem ZMod.euler_criterion (p : ) [Fact ()] {a : ZMod p} (ha : a 0) :
a ^ (p / 2) = 1

Euler's Criterion: a nonzero a : ZMod p is a square if and only if x ^ (p / 2) = 1.

theorem ZMod.pow_div_two_eq_neg_one_or_one (p : ) [Fact ()] {a : ZMod p} (ha : a 0) :
a ^ (p / 2) = 1 a ^ (p / 2) = -1

If a : ZMod p is nonzero, then a^(p/2) is either 1 or -1.

### Definition of the Legendre symbol and basic properties #

def legendreSym (p : ) [Fact ()] (a : ) :

The Legendre symbol of a : ℤ and a prime p, legendreSym p a, is an integer defined as

• 0 if a is 0 modulo p;
• 1 if a is a nonzero square modulo p
• -1 otherwise.

Note the order of the arguments! The advantage of the order chosen here is that legendreSym p is a multiplicative function ℤ → ℤ.

Equations
• = () a
Instances For
theorem legendreSym.eq_pow (p : ) [Fact ()] (a : ) :
() = a ^ (p / 2)

We have the congruence legendreSym p a ≡ a ^ (p / 2) mod p.

theorem legendreSym.eq_one_or_neg_one (p : ) [Fact ()] {a : } (ha : a 0) :
= 1 = -1

If p ∤ a, then legendreSym p a is 1 or -1.

theorem legendreSym.eq_neg_one_iff_not_one (p : ) [Fact ()] {a : } (ha : a 0) :
= -1 ¬ = 1
theorem legendreSym.eq_zero_iff (p : ) [Fact ()] (a : ) :
= 0 a = 0

The Legendre symbol of p and a is zero iff p ∣ a.

@[simp]
theorem legendreSym.at_zero (p : ) [Fact ()] :
= 0
@[simp]
theorem legendreSym.at_one (p : ) [Fact ()] :
= 1
theorem legendreSym.mul (p : ) [Fact ()] (a : ) (b : ) :
legendreSym p (a * b) = *

The Legendre symbol is multiplicative in a for p fixed.

@[simp]
theorem legendreSym.hom_apply (p : ) [Fact ()] (a : ) :
() a =
def legendreSym.hom (p : ) [Fact ()] :

The Legendre symbol is a homomorphism of monoids with zero.

Equations
• = { toFun := , map_zero' := , map_one' := , map_mul' := }
Instances For
theorem legendreSym.sq_one (p : ) [Fact ()] {a : } (ha : a 0) :
^ 2 = 1

The square of the symbol is 1 if p ∤ a.

theorem legendreSym.sq_one' (p : ) [Fact ()] {a : } (ha : a 0) :
legendreSym p (a ^ 2) = 1

The Legendre symbol of a^2 at p is 1 if p ∤ a.

theorem legendreSym.mod (p : ) [Fact ()] (a : ) :
= legendreSym p (a % p)

The Legendre symbol depends only on a mod p.

theorem legendreSym.eq_one_iff (p : ) [Fact ()] {a : } (ha0 : a 0) :
= 1 IsSquare a

When p ∤ a, then legendreSym p a = 1 iff a is a square mod p.

theorem legendreSym.eq_one_iff' (p : ) [Fact ()] {a : } (ha0 : a 0) :
legendreSym p a = 1 IsSquare a
theorem legendreSym.eq_neg_one_iff (p : ) [Fact ()] {a : } :
= -1 ¬IsSquare a

legendreSym p a = -1 iff a is a nonsquare mod p.

theorem legendreSym.eq_neg_one_iff' (p : ) [Fact ()] {a : } :
legendreSym p a = -1 ¬IsSquare a
theorem legendreSym.card_sqrts (p : ) [Fact ()] (hp : p 2) (a : ) :
{x : ZMod p | x ^ 2 = a}.toFinset.card = + 1

The number of square roots of a modulo p is determined by the Legendre symbol.

### Applications to binary quadratic forms #

theorem legendreSym.eq_one_of_sq_sub_mul_sq_eq_zero {p : } [Fact ()] {a : } (ha : a 0) {x : ZMod p} {y : ZMod p} (hy : y 0) (hxy : x ^ 2 - a * y ^ 2 = 0) :
= 1

The Legendre symbol legendreSym p a = 1 if there is a solution in ℤ/pℤ of the equation x^2 - a*y^2 = 0 with y ≠ 0.

theorem legendreSym.eq_one_of_sq_sub_mul_sq_eq_zero' {p : } [Fact ()] {a : } (ha : a 0) {x : ZMod p} {y : ZMod p} (hx : x 0) (hxy : x ^ 2 - a * y ^ 2 = 0) :
= 1

The Legendre symbol legendreSym p a = 1 if there is a solution in ℤ/pℤ of the equation x^2 - a*y^2 = 0 with x ≠ 0.

theorem legendreSym.eq_zero_mod_of_eq_neg_one {p : } [Fact ()] {a : } (h : = -1) {x : ZMod p} {y : ZMod p} (hxy : x ^ 2 - a * y ^ 2 = 0) :
x = 0 y = 0

If legendreSym p a = -1, then the only solution of x^2 - a*y^2 = 0 in ℤ/pℤ is the trivial one.

theorem legendreSym.prime_dvd_of_eq_neg_one {p : } [Fact ()] {a : } (h : = -1) {x : } {y : } (hxy : p x ^ 2 - a * y ^ 2) :
p x p y

If legendreSym p a = -1 and p divides x^2 - a*y^2, then p must divide x and y.

### The value of the Legendre symbol at -1#

See jacobiSym.at_neg_one for the corresponding statement for the Jacobi symbol.

theorem legendreSym.at_neg_one {p : } [Fact ()] (hp : p 2) :
legendreSym p (-1) = ZMod.χ₄ p

legendreSym p (-1) is given by χ₄ p.

theorem ZMod.exists_sq_eq_neg_one_iff {p : } [Fact ()] :
IsSquare (-1) p % 4 3

-1 is a square in ZMod p iff p is not congruent to 3 mod 4.

theorem ZMod.mod_four_ne_three_of_sq_eq_neg_one {p : } [Fact ()] {y : ZMod p} (hy : y ^ 2 = -1) :
p % 4 3
theorem ZMod.mod_four_ne_three_of_sq_eq_neg_sq' {p : } [Fact ()] {x : ZMod p} {y : ZMod p} (hy : y 0) (hxy : x ^ 2 = -y ^ 2) :
p % 4 3

If two nonzero squares are negatives of each other in ZMod p, then p % 4 ≠ 3.

theorem ZMod.mod_four_ne_three_of_sq_eq_neg_sq {p : } [Fact ()] {x : ZMod p} {y : ZMod p} (hx : x 0) (hxy : x ^ 2 = -y ^ 2) :
p % 4 3