Quadratic characters on ℤ/nℤ

This file defines some quadratic characters on the rings ℤ/4ℤ and ℤ/8ℤ.

We set them up to be of type MulChar (ZMod n) ℤ, where n is 4 or 8.

Tags

quadratic character, zmod

Quadratic characters mod 4 and 8

We define the primitive quadratic characters χ₄ on ZMod 4 and χ₈, χ₈' on ZMod 8.

def ZMod.χ₄ : MulChar (ZMod 4) ℤ

Define the nontrivial quadratic character on ZMod 4, χ₄. It corresponds to the extension ℚ(√-1)/ℚ.

@[simp]

theorem ZMod.χ₄_apply (a : ZMod 4) : χ₄ a = match a with | 0 => 0 | 2 => 0 | 1 => 1 | 3 => -1

theorem ZMod.isQuadratic_χ₄ : χ₄.IsQuadratic

χ₄ takes values in {0, 1, -1}

theorem ZMod.χ₄_nat_mod_four (n : ℕ) : χ₄ ↑n = χ₄ ↑(n % 4)

The value of χ₄ n, for n : ℕ, depends only on n % 4.

theorem ZMod.χ₄_int_mod_four (n : ℤ) : χ₄ ↑n = χ₄ ↑(n % 4)

The value of χ₄ n, for n : ℤ, depends only on n % 4.

theorem ZMod.χ₄_int_eq_if_mod_four (n : ℤ) : χ₄ ↑n = if n % 2 = 0 then 0 else if n % 4 = 1 then 1 else -1

An explicit description of χ₄ on integers / naturals

theorem ZMod.χ₄_nat_eq_if_mod_four (n : ℕ) : χ₄ ↑n = if n % 2 = 0 then 0 else if n % 4 = 1 then 1 else -1

theorem ZMod.χ₄_eq_neg_one_pow {n : ℕ} (hn : n % 2 = 1) : χ₄ ↑n = (-1) ^ (n / 2)

Alternative description of χ₄ n for odd n : ℕ in terms of powers of -1

theorem ZMod.χ₄_nat_one_mod_four {n : ℕ} (hn : n % 4 = 1) : χ₄ ↑n = 1

If n % 4 = 1, then χ₄ n = 1.

theorem ZMod.χ₄_nat_three_mod_four {n : ℕ} (hn : n % 4 = 3) : χ₄ ↑n = -1

If n % 4 = 3, then χ₄ n = -1.

theorem ZMod.χ₄_int_one_mod_four {n : ℤ} (hn : n % 4 = 1) : χ₄ ↑n = 1

If n % 4 = 1, then χ₄ n = 1.

theorem ZMod.χ₄_int_three_mod_four {n : ℤ} (hn : n % 4 = 3) : χ₄ ↑n = -1

If n % 4 = 3, then χ₄ n = -1.

theorem ZMod.neg_one_pow_div_two_of_one_mod_four {n : ℕ} (hn : n % 4 = 1) : (-1) ^ (n / 2) = 1

If n % 4 = 1, then (-1)^(n/2) = 1.

theorem ZMod.neg_one_pow_div_two_of_three_mod_four {n : ℕ} (hn : n % 4 = 3) : (-1) ^ (n / 2) = -1

If n % 4 = 3, then (-1)^(n/2) = -1.

def ZMod.χ₈ : MulChar (ZMod 8) ℤ

Define the first primitive quadratic character on ZMod 8, χ₈. It corresponds to the extension ℚ(√2)/ℚ.

@[simp]

theorem ZMod.χ₈_apply (a : ZMod 8) : χ₈ a = match a with | 0 => 0 | 2 => 0 | 4 => 0 | 6 => 0 | 1 => 1 | 7 => 1 | 3 => -1 | 5 => -1

theorem ZMod.isQuadratic_χ₈ : χ₈.IsQuadratic

χ₈ takes values in {0, 1, -1}

theorem ZMod.χ₈_nat_mod_eight (n : ℕ) : χ₈ ↑n = χ₈ ↑(n % 8)

The value of χ₈ n, for n : ℕ, depends only on n % 8.

theorem ZMod.χ₈_int_mod_eight (n : ℤ) : χ₈ ↑n = χ₈ ↑(n % 8)

The value of χ₈ n, for n : ℤ, depends only on n % 8.

theorem ZMod.χ₈_int_eq_if_mod_eight (n : ℤ) : χ₈ ↑n = if n % 2 = 0 then 0 else if n % 8 = 1 ∨ n % 8 = 7 then 1 else -1

An explicit description of χ₈ on integers / naturals

theorem ZMod.χ₈_nat_eq_if_mod_eight (n : ℕ) : χ₈ ↑n = if n % 2 = 0 then 0 else if n % 8 = 1 ∨ n % 8 = 7 then 1 else -1

def ZMod.χ₈' : MulChar (ZMod 8) ℤ

Define the second primitive quadratic character on ZMod 8, χ₈'. It corresponds to the extension ℚ(√-2)/ℚ.

@[simp]

theorem ZMod.χ₈'_apply (a : ZMod 8) : χ₈' a = match a with | 0 => 0 | 2 => 0 | 4 => 0 | 6 => 0 | 1 => 1 | 3 => 1 | 5 => -1 | 7 => -1

theorem ZMod.isQuadratic_χ₈' : χ₈'.IsQuadratic

χ₈' takes values in {0, 1, -1}

theorem ZMod.χ₈'_int_eq_if_mod_eight (n : ℤ) : χ₈' ↑n = if n % 2 = 0 then 0 else if n % 8 = 1 ∨ n % 8 = 3 then 1 else -1

An explicit description of χ₈' on integers / naturals

theorem ZMod.χ₈'_nat_eq_if_mod_eight (n : ℕ) : χ₈' ↑n = if n % 2 = 0 then 0 else if n % 8 = 1 ∨ n % 8 = 3 then 1 else -1

theorem ZMod.χ₈'_eq_χ₄_mul_χ₈ (a : ZMod 8) : χ₈' a = χ₄ a.cast * χ₈ a

The relation between χ₄, χ₈ and χ₈'

theorem ZMod.χ₈'_int_eq_χ₄_mul_χ₈ (a : ℤ) : χ₈' ↑a = χ₄ ↑a * χ₈ ↑a