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Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic

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 quadraticCharFun (α : Type u_1) [MonoidWithZero α] [DecidableEq α] [DecidablePred IsSquare] (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 quadraticCharFun_mul.

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

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    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 quadraticCharFun_eq_zero_iff {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] {a : F} :

    Some basic API lemmas

    @[simp]
    @[simp]
    theorem quadraticCharFun_one {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] :
    theorem quadraticCharFun_eq_one_of_char_two {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] (hF : ringChar F = 2) {a : F} (ha : a 0) :

    If ringChar F = 2, then quadraticCharFun F takes the value 1 on nonzero elements.

    theorem quadraticCharFun_eq_pow_of_char_ne_two {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] (hF : ringChar F 2) {a : F} (ha : a 0) :
    quadraticCharFun F a = if a ^ (Fintype.card F / 2) = 1 then 1 else -1

    If ringChar F is odd, then quadraticCharFun F a can be computed in terms of a ^ (Fintype.card F / 2).

    theorem quadraticCharFun_mul {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] (a : F) (b : F) :

    The quadratic character is multiplicative.

    @[simp]
    theorem quadraticChar_apply (F : Type u_1) [Field F] [Fintype F] [DecidableEq F] (a : F) :
    def quadraticChar (F : Type u_1) [Field F] [Fintype F] [DecidableEq F] :

    The quadratic character as a multiplicative character.

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      theorem quadraticChar_eq_zero_iff {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] {a : F} :
      (quadraticChar F) a = 0 a = 0

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

      theorem quadraticChar_zero {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] :
      theorem quadraticChar_one_iff_isSquare {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] {a : F} (ha : a 0) :

      For nonzero a : F, quadraticChar F a = 1 ↔ IsSquare a.

      theorem quadraticChar_sq_one' {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] {a : F} (ha : a 0) :
      (quadraticChar F) (a ^ 2) = 1

      The quadratic character takes the value 1 on nonzero squares.

      theorem quadraticChar_sq_one {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] {a : F} (ha : a 0) :
      (quadraticChar F) a ^ 2 = 1

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

      theorem quadraticChar_dichotomy {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] {a : F} (ha : a 0) :
      (quadraticChar F) a = 1 (quadraticChar F) a = -1

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

      theorem quadraticChar_eq_neg_one_iff_not_one {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] {a : F} (ha : a 0) :

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

      For a : F, quadraticChar F a = -1 ↔ ¬ IsSquare a.

      theorem quadraticChar_exists_neg_one {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] (hF : ringChar F 2) :
      ∃ (a : F), (quadraticChar F) a = -1

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

      theorem quadraticChar_eq_one_of_char_two {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] (hF : ringChar F = 2) {a : F} (ha : a 0) :

      If ringChar F = 2, then quadraticChar F takes the value 1 on nonzero elements.

      theorem quadraticChar_eq_pow_of_char_ne_two {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] (hF : ringChar F 2) {a : F} (ha : a 0) :
      (quadraticChar F) a = if a ^ (Fintype.card F / 2) = 1 then 1 else -1

      If ringChar F is odd, then quadraticChar F a can be computed in terms of a ^ (Fintype.card F / 2).

      theorem quadraticChar_eq_pow_of_char_ne_two' {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] (hF : ringChar F 2) (a : F) :
      ((quadraticChar F) a) = a ^ (Fintype.card F / 2)
      theorem quadraticChar_isQuadratic (F : Type u_1) [Field F] [Fintype F] [DecidableEq F] :
      (quadraticChar F).IsQuadratic

      The quadratic character is quadratic as a multiplicative character.

      theorem quadraticChar_isNontrivial {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] (hF : ringChar F 2) :
      (quadraticChar F).IsNontrivial

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

      theorem quadraticChar_card_sqrts {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] (hF : ringChar F 2) (a : F) :
      {x : F | x ^ 2 = a}.toFinset.card = (quadraticChar F) a + 1

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

      theorem quadraticChar_sum_zero {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] (hF : ringChar F 2) :
      a : F, (quadraticChar 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 quadraticChar F (-1) in terms of χ₄.

      theorem quadraticChar_neg_one {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] (hF : ringChar F 2) :
      (quadraticChar F) (-1) = ZMod.χ₄ (Fintype.card F)

      The value of the quadratic character at -1

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