Quadratic characters of finite fields #

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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.

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def quadratic_char_fun (α : Type u_1) [monoid_with_zero α] [decidable_eq α] [decidable_pred is_square] (a : α) :

ℤ

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_fun_mul.

We will later define quadratic_char to be a multiplicative character of type mul_char F ℤ, when the domain is a finite field F.

Equations

quadratic_char_fun 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.

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theorem quadratic_char_fun_eq_zero_iff {F : Type u_1} [field F] [fintype F] [decidable_eq F] {a : F} :

quadratic_char_fun F a = 0 ↔ a = 0

Some basic API lemmas

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@[simp]

theorem quadratic_char_fun_zero {F : Type u_1} [field F] [fintype F] [decidable_eq F] :

quadratic_char_fun F 0 = 0

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@[simp]

theorem quadratic_char_fun_one {F : Type u_1} [field F] [fintype F] [decidable_eq F] :

quadratic_char_fun F 1 = 1

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theorem quadratic_char_fun_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) :

quadratic_char_fun F a = 1

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

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theorem quadratic_char_fun_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) :

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

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

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

quadratic_char_fun F (a * b) = quadratic_char_fun F a * quadratic_char_fun F b

The quadratic character is multiplicative.

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def quadratic_char (F : Type u_1) [field F] [fintype F] [decidable_eq F] :

mul_char F ℤ

The quadratic character as a multiplicative character.

Equations

quadratic_char F = {to_monoid_hom := {to_fun := quadratic_char_fun F (λ (a : F), fintype.is_square.decidable_pred a), map_one' := _, map_mul' := _}, map_nonunit' := _}

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@[simp]

theorem quadratic_char_apply (F : Type u_1) [field F] [fintype F] [decidable_eq F] (a : F) :

⇑(quadratic_char F) a = quadratic_char_fun F a

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theorem quadratic_char_eq_zero_iff {F : Type u_1} [field F] [fintype F] [decidable_eq F] {a : F} :

⇑(quadratic_char F) a = 0 ↔ a = 0

The value of the quadratic character on a is zero iff a = 0.

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@[simp]

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

⇑(quadratic_char F) 0 = 0

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

⇑(quadratic_char F) a = 1 ↔ is_square a

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

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

⇑(quadratic_char F) (a ^ 2) = 1

The quadratic character takes the value 1 on nonzero squares.

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

⇑(quadratic_char F) a ^ 2 = 1

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

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

⇑(quadratic_char F) a = 1 ∨ ⇑(quadratic_char F) a = -1

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

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

⇑(quadratic_char F) a = -1 ↔ ¬⇑(quadratic_char F) a = 1

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

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

⇑(quadratic_char F) a = -1 ↔ ¬is_square a

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

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

∃ (a : F), ⇑(quadratic_char F) a = -1

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

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theorem 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) :

⇑(quadratic_char F) a = 1

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

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theorem 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) :

⇑(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).

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theorem 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) :

↑(⇑(quadratic_char F) a) = a ^ (fintype.card F / 2)

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theorem quadratic_char_is_quadratic (F : Type u_1) [field F] [fintype F] [decidable_eq F] :

(quadratic_char F).is_quadratic

The quadratic character is quadratic as a multiplicative character.

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

(quadratic_char F).is_nontrivial

The quadratic character is nontrivial as a multiplicative character when the domain has odd characteristic.

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theorem 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) = ⇑(quadratic_char F) a + 1

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

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theorem 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), ⇑(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.

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

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

The value of the quadratic character at -1

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

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

-1 is a square in F iff #F is not congruent to 3 mod 4.