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number_theory.legendre_symbol.quadratic_char

Quadratic characters of finite fields #

This file defines the quadratic character on a finite field F and proves some basic statements about it.

Tags #

quadratic character

Definition of the quadratic character #

We define the quadratic character of a finite field F with values in ℤ.

def char.quadratic_char (α : Type u_1) [monoid_with_zero α] [decidable_eq α] [decidable_pred is_square] (a : α) :

Define the quadratic character with values in ℤ on a monoid with zero α. It takes the value zero at zero; for non-zero argument a : α, it is 1 if a is a square, otherwise it is -1.

This only deserves the name "character" when it is multiplicative, e.g., when α is a finite field. See quadratic_char_mul.

Equations

char.quadratic_char α a = ite (a = 0) 0 (ite (is_square a) 1 (-1))

Basic properties of the quadratic character #

We prove some properties of the quadratic character. We work with a finite field F here. The interesting case is when the characteristic of F is odd.

theorem char.quadratic_char_eq_zero_iff {F : Type u_1} [field F] [fintype F] [decidable_eq F] (a : F) :

char.quadratic_char F a = 0 ↔ a = 0

Some basic API lemmas

@[simp]

theorem char.quadratic_char_zero {F : Type u_1} [field F] [fintype F] [decidable_eq F] :

char.quadratic_char F 0 = 0

@[simp]

theorem char.quadratic_char_one {F : Type u_1} [field F] [fintype F] [decidable_eq F] :

char.quadratic_char F 1 = 1

theorem char.quadratic_char_one_iff_is_square {F : Type u_1} [field F] [fintype F] [decidable_eq F] {a : F} (ha : a ≠ 0) :

char.quadratic_char F a = 1 ↔ is_square a

For nonzero a : F, quadratic_char F a = 1 ↔ is_square a.

theorem char.quadratic_char_sq_one' {F : Type u_1} [field F] [fintype F] [decidable_eq F] {a : F} (ha : a ≠ 0) :

char.quadratic_char F (a ^ 2) = 1

The quadratic character takes the value 1 on nonzero squares.

theorem char.quadratic_char_eq_one_of_char_two {F : Type u_1} [field F] [fintype F] [decidable_eq F] (hF : ring_char F = 2) {a : F} (ha : a ≠ 0) :

char.quadratic_char F a = 1

If ring_char F = 2, then quadratic_char F takes the value 1 on nonzero elements.

theorem char.quadratic_char_eq_pow_of_char_ne_two {F : Type u_1} [field F] [fintype F] [decidable_eq F] (hF : ring_char F ≠ 2) {a : F} (ha : a ≠ 0) :

char.quadratic_char F a = ite (a ^ (fintype.card F / 2) = 1) 1 (-1)

If ring_char F is odd, then quadratic_char F a can be computed in terms of a ^ (fintype.card F / 2).

theorem char.quadratic_char_mul {F : Type u_1} [field F] [fintype F] [decidable_eq F] (a b : F) :

char.quadratic_char F (a * b) = char.quadratic_char F a * char.quadratic_char F b

The quadratic character is multiplicative.

@[simp]

theorem char.quadratic_char_hom_apply {F : Type u_1} [field F] [fintype F] [decidable_eq F] (a : F) :

⇑char.quadratic_char_hom a = char.quadratic_char F a

def char.quadratic_char_hom {F : Type u_1} [field F] [fintype F] [decidable_eq F] :

The quadratic character is a homomorphism of monoids with zero.

Equations

char.quadratic_char_hom = {to_fun := char.quadratic_char F (λ (a : F), is_square_decidable a), map_zero' := _, map_one' := _, map_mul' := _}

theorem char.quadratic_char_sq_one {F : Type u_1} [field F] [fintype F] [decidable_eq F] {a : F} (ha : a ≠ 0) :

char.quadratic_char F a ^ 2 = 1

The square of the quadratic character on nonzero arguments is 1.

theorem char.quadratic_char_dichotomy {F : Type u_1} [field F] [fintype F] [decidable_eq F] {a : F} (ha : a ≠ 0) :

char.quadratic_char F a = 1 ∨ char.quadratic_char F a = -1

The quadratic character is 1 or -1 on nonzero arguments.

theorem char.quadratic_char_eq_neg_one_iff_not_one {F : Type u_1} [field F] [fintype F] [decidable_eq F] {a : F} (ha : a ≠ 0) :

char.quadratic_char F a = -1 ↔ ¬char.quadratic_char F a = 1

A variant

theorem char.quadratic_char_neg_one_iff_not_is_square {F : Type u_1} [field F] [fintype F] [decidable_eq F] {a : F} :

char.quadratic_char F a = -1 ↔ ¬is_square a

For a : F, quadratic_char F a = -1 ↔ ¬ is_square a.

theorem char.quadratic_char_exists_neg_one {F : Type u_1} [field F] [fintype F] [decidable_eq F] (hF : ring_char F ≠ 2) :

∃ (a : F), char.quadratic_char F a = -1

If F has odd characteristic, then quadratic_char F takes the value -1.

theorem char.quadratic_char_card_sqrts {F : Type u_1} [field F] [fintype F] [decidable_eq F] (hF : ring_char F ≠ 2) (a : F) :

↑({x : F | x ^ 2 = a}.to_finset.card) = char.quadratic_char F a + 1

The number of solutions to x^2 = a is determined by the quadratic character.

theorem char.quadratic_char_sum_zero {F : Type u_1} [field F] [fintype F] [decidable_eq F] (hF : ring_char F ≠ 2) :

finset.univ.sum (λ (a : F), char.quadratic_char F a) = 0

The sum over the values of the quadratic character is zero when the characteristic is odd.

Special values of the quadratic character #

We express quadratic_char F (-1) in terms of χ₄.

theorem char.quadratic_char_neg_one {F : Type u_1} [field F] [fintype F] [decidable_eq F] (hF : ring_char F ≠ 2) :

char.quadratic_char F (-1) = ⇑zmod.χ₄ ↑(fintype.card F)

The value of the quadratic character at -1

theorem char.is_square_neg_one_iff {F : Type u_1} [field F] [fintype F] :

is_square (-1) ↔ fintype.card F % 4 ≠ 3

The interpretation in terms of whether -1 is a square in F