Quadratic characters of finite fields

Further facts relying on Gauss sums.

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 quadraticChar_two {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] (hF : ringChar F ≠ 2) : (quadraticChar F) 2 = ZMod.χ₈ ↑(Fintype.card F)

The value of the quadratic character at 2

theorem FiniteField.isSquare_two_iff {F : Type u_1} [Field F] [Fintype F] : IsSquare 2 ↔ Fintype.card F % 8 ≠ 3 ∧ Fintype.card F % 8 ≠ 5

2 is a square in F iff #F is not congruent to 3 or 5 mod 8.

theorem quadraticChar_neg_two {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] (hF : ringChar F ≠ 2) : (quadraticChar F) (-2) = ZMod.χ₈' ↑(Fintype.card F)

The value of the quadratic character at -2

theorem FiniteField.isSquare_neg_two_iff {F : Type u_1} [Field F] [Fintype F] : IsSquare (-2) ↔ Fintype.card F % 8 ≠ 5 ∧ Fintype.card F % 8 ≠ 7

-2 is a square in F iff #F is not congruent to 5 or 7 mod 8.

theorem quadraticChar_card_card {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] (hF : ringChar F ≠ 2) {F' : Type u_2} [Field F'] [Fintype F'] [DecidableEq F'] (hF' : ringChar F' ≠ 2) (h : ringChar F' ≠ ringChar F) : (quadraticChar F) ↑(Fintype.card F') = (quadraticChar F') (↑((quadraticChar F) (-1)) * ↑(Fintype.card F))

The relation between the values of the quadratic character of one field F at the cardinality of another field F' and of the quadratic character of F' at the cardinality of F.

theorem quadraticChar_odd_prime {F : Type u_1} [Field F] [Fintype F] [DecidableEq F] (hF : ringChar F ≠ 2) {p : ℕ} [Fact (Nat.Prime p)] (hp₁ : p ≠ 2) (hp₂ : ringChar F ≠ p) : (quadraticChar F) ↑p = (quadraticChar (ZMod p)) (↑(ZMod.χ₄ ↑(Fintype.card F)) * ↑(Fintype.card F))

The value of the quadratic character at an odd prime p different from ringChar F.

theorem FiniteField.isSquare_odd_prime_iff {F : Type u_1} [Field F] [Fintype F] (hF : ringChar F ≠ 2) {p : ℕ} [Fact (Nat.Prime p)] (hp : p ≠ 2) : IsSquare ↑p ↔ (quadraticChar (ZMod p)) (↑(ZMod.χ₄ ↑(Fintype.card F)) * ↑(Fintype.card F)) ≠ -1

An odd prime p is a square in F iff the quadratic character of ZMod p does not take the value -1 on χ₄#F * #F.