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Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum

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

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

-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 : } [Fact p.Prime] (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 : } [Fact p.Prime] (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.