Gauss sums #

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We define the Gauss sum associated to a multiplicative and an additive character of a finite field and prove some results about them.

Main definition #

Let R be a finite commutative ring and let R' be another commutative ring. If χ is a multiplicative character R → R' (type mul_char R R') and ψ is an additive character R → R' (type add_char R R', which abbreviates (multiplicative R) →* R'), then the Gauss sum of χ and ψ is ∑ a, χ a * ψ a.

Main results #

Some important results are as follows.

gauss_sum_mul_gauss_sum_eq_card: The product of the Gauss sums of χ and ψ and that of χ⁻¹ and ψ⁻¹ is the cardinality of the source ring R (if χ is nontrivial, ψ is primitive and R is a field).
gauss_sum_sq: The square of the Gauss sum is χ(-1) times the cardinality of R if in addition χ is a quadratic character.
quad_gauss_sum_frob: For a quadratic character χ, raising the Gauss sum to the pth power (where p is the characteristic of the target ring R') multiplies it by χ p.
char.card_pow_card: When F and F' are finite fields and χ : F → F' is a nontrivial quadratic character, then (χ (-1) * #F)^(#F'/2) = χ (#F').
finite_field.two_pow_card: For every finite field F of odd characteristic, we have 2^(#F/2) = χ₈(#F) in F.

This machinery can be used to derive (a generalization of) the Law of Quadratic Reciprocity.

Tags #

additive character, multiplicative character, Gauss sum

Definition and first properties #

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def gauss_sum {R : Type u} [comm_ring R] [fintype R] {R' : Type v} [comm_ring R'] (χ : mul_char R R') (ψ : add_char R R') :

Definition of the Gauss sum associated to a multiplicative and an additive character.

Equations

gauss_sum χ ψ = finset.univ.sum (λ (a : R), ⇑χ a * ⇑ψ a)

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theorem gauss_sum_mul_shift {R : Type u} [comm_ring R] [fintype R] {R' : Type v} [comm_ring R'] (χ : mul_char R R') (ψ : add_char R R') (a : Rˣ) :

⇑χ ↑a * gauss_sum χ (ψ.mul_shift ↑a) = gauss_sum χ ψ

Replacing ψ by mul_shift ψ a and multiplying the Gauss sum by χ a does not change it.

The product of two Gauss sums #

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theorem gauss_sum_mul_gauss_sum_eq_card {R : Type u} [field R] [fintype R] {R' : Type v} [comm_ring R'] [is_domain R'] {χ : mul_char R R'} (hχ : χ.is_nontrivial) {ψ : add_char R R'} (hψ : ψ.is_primitive) :

gauss_sum χ ψ * gauss_sum χ⁻¹ ψ⁻¹ = ↑(fintype.card R)

We have gauss_sum χ ψ * gauss_sum χ⁻¹ ψ⁻¹ = fintype.card R when χ is nontrivial and ψ is primitive (and R is a field).

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theorem gauss_sum_sq {R : Type u} [field R] [fintype R] {R' : Type v} [comm_ring R'] [is_domain R'] {χ : mul_char R R'} (hχ₁ : χ.is_nontrivial) (hχ₂ : χ.is_quadratic) {ψ : add_char R R'} (hψ : ψ.is_primitive) :

gauss_sum χ ψ ^ 2 = ⇑χ (-1) * ↑(fintype.card R)

When χ is a nontrivial quadratic character, then the square of gauss_sum χ ψ is χ(-1) times the cardinality of R.

Gauss sums and Frobenius #

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theorem gauss_sum_frob {R : Type u} [comm_ring R] [fintype R] {R' : Type v} [comm_ring R'] (p : ℕ) [fp : fact (nat.prime p)] [hch : char_p R' p] (χ : mul_char R R') (ψ : add_char R R') :

gauss_sum χ ψ ^ p = gauss_sum (χ ^ p) (ψ ^ p)

When R' has prime characteristic p, then the pth power of the Gauss sum of χ and ψ is the Gauss sum of χ^p and ψ^p.

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theorem mul_char.is_quadratic.gauss_sum_frob {R : Type u} [comm_ring R] [fintype R] {R' : Type v} [comm_ring R'] (p : ℕ) [fp : fact (nat.prime p)] [hch : char_p R' p] (hp : is_unit ↑p) {χ : mul_char R R'} (hχ : χ.is_quadratic) (ψ : add_char R R') :

gauss_sum χ ψ ^ p = ⇑χ ↑p * gauss_sum χ ψ

For a quadratic character χ and when the characteristic p of the target ring is a unit in the source ring, the pth power of the Gauss sum ofχ and ψ is χ p times the original Gauss sum.

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theorem mul_char.is_quadratic.gauss_sum_frob_iter {R : Type u} [comm_ring R] [fintype R] {R' : Type v} [comm_ring R'] (p : ℕ) [fp : fact (nat.prime p)] [hch : char_p R' p] (n : ℕ) (hp : is_unit ↑p) {χ : mul_char R R'} (hχ : χ.is_quadratic) (ψ : add_char R R') :

gauss_sum χ ψ ^ p ^ n = ⇑χ (↑p ^ n) * gauss_sum χ ψ

For a quadratic character χ and when the characteristic p of the target ring is a unit in the source ring and n is a natural number, the p^nth power of the Gauss sum ofχ and ψ is χ (p^n) times the original Gauss sum.

Values of quadratic characters #

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theorem char.card_pow_char_pow {R : Type u} [comm_ring R] [fintype R] {R' : Type v} [comm_ring R'] [is_domain R'] {χ : mul_char R R'} (hχ : χ.is_quadratic) (ψ : add_char R R') (p n : ℕ) [fp : fact (nat.prime p)] [hch : char_p R' p] (hp : is_unit ↑p) (hp' : p ≠ 2) (hg : gauss_sum χ ψ ^ 2 = ⇑χ (-1) * ↑(fintype.card R)) :

(⇑χ (-1) * ↑(fintype.card R)) ^ (p ^ n / 2) = ⇑χ (↑p ^ n)

If the square of the Gauss sum of a quadratic character is χ(-1) * #R, then we get, for all n : ℕ, the relation (χ(-1) * #R) ^ (p^n/2) = χ(p^n), where p is the (odd) characteristic of the target ring R'. This version can be used when R is not a field, e.g., ℤ/8ℤ.

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theorem char.card_pow_card {F : Type u_1} [field F] [fintype F] {F' : Type u_2} [field F'] [fintype F'] {χ : mul_char F F'} (hχ₁ : χ.is_nontrivial) (hχ₂ : χ.is_quadratic) (hch₁ : ring_char F' ≠ ring_char F) (hch₂ : ring_char F' ≠ 2) :

(⇑χ (-1) * ↑(fintype.card F)) ^ (fintype.card F' / 2) = ⇑χ ↑(fintype.card F')

When F and F' are finite fields and χ : F → F' is a nontrivial quadratic character, then (χ(-1) * #F)^(#F'/2) = χ(#F').

The quadratic character of 2 #

This section proves the following result.

For every finite field F of odd characteristic, we have 2^(#F/2) = χ₈(#F) in F. This can be used to show that the quadratic character of F takes the value χ₈(#F) at 2.

The proof uses the Gauss sum of χ₈ and a primitive additive character on ℤ/8ℤ; in this way, the result is reduced to card_pow_char_pow.

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theorem finite_field.two_pow_card {F : Type u_1} [fintype F] [field F] (hF : ring_char F ≠ 2) :

2 ^ (fintype.card F / 2) = ↑(⇑zmod.χ₈ ↑(fintype.card F))

For every finite field F of odd characteristic, we have 2^(#F/2) = χ₈(#F) in F.