Finite fields

This file contains basic results about finite fields. Throughout most of this file, K denotes a finite field and q is notation for the cardinality of K.

Main results

Every finite integral domain is a field (field_of_integral_domain).
The unit group of a finite field is a cyclic group of order q - 1. (finite_field.is_cyclic and card_units)
sum_pow_units: The sum of x^i, where x ranges over the units of K, is
- q-1 if q-1 ∣ i
- 0 otherwise
finite_field.card: The cardinality q is a power of the characteristic of K. See card' for a variant.

Notation

Throughout most of this file, K denotes a finite field and q is notation for the cardinality of K.

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theorem finite_field.card_image_polynomial_eval {R : Type u_2} [integral_domain R] [decidable_eq R] [fintype R] {p : polynomial R} (hp : 0 < p.degree) :

fintype.card R ≤ (p.nat_degree) * (finset.image (λ (x : R), polynomial.eval x p) finset.univ).card

The cardinality of a field is at most n times the cardinality of the image of a degree n polynomial

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theorem finite_field.exists_root_sum_quadratic {R : Type u_2} [integral_domain R] [fintype R] {f g : polynomial R} (hf2 : f.degree = 2) (hg2 : g.degree = 2) (hR : fintype.card R % 2 = 1) :

∃ (a b : R), polynomial.eval a f + polynomial.eval b g = 0

If f and g are quadratic polynomials, then the f.eval a + g.eval b = 0 has a solution.

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theorem finite_field.card_units {K : Type u_1} [field K] [fintype K] :

fintype.card (units K) = fintype.card K - 1

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theorem finite_field.prod_univ_units_id_eq_neg_one {K : Type u_1} [field K] [fintype K] :

∏ (x : units K), x = -1

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theorem finite_field.pow_card_sub_one_eq_one {K : Type u_1} [field K] [fintype K] (a : K) (ha : a ≠ 0) :

a ^ (fintype.card K - 1) = 1

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theorem finite_field.pow_card {K : Type u_1} [field K] [fintype K] (a : K) :

a ^ fintype.card K = a

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theorem finite_field.card (K : Type u_1) [field K] [fintype K] (p : ℕ) [char_p K p] :

∃ (n : ℕ+), nat.prime p ∧ fintype.card K = p ^ ↑n

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theorem finite_field.card' {K : Type u_1} [field K] [fintype K] :

∃ (p : ℕ) (n : ℕ+), nat.prime p ∧ fintype.card K = p ^ ↑n

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@[simp]

theorem finite_field.cast_card_eq_zero (K : Type u_1) [field K] [fintype K] :

↑(fintype.card K) = 0

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theorem finite_field.forall_pow_eq_one_iff (K : Type u_1) [field K] [fintype K] (i : ℕ) :

(∀ (x : units K), x ^ i = 1) ↔ fintype.card K - 1 ∣ i

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theorem finite_field.sum_pow_units (K : Type u_1) [field K] [fintype K] (i : ℕ) :

∑ (x : units K), ↑x ^ i = ite (fintype.card K - 1 ∣ i) (-1) 0

The sum of x ^ i as x ranges over the units of a finite field of cardinality q is equal to 0 unless (q - 1) ∣ i, in which case the sum is q - 1.

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theorem finite_field.sum_pow_lt_card_sub_one {K : Type u_1} [field K] [fintype K] (i : ℕ) (h : i < fintype.card K - 1) :

∑ (x : K), x ^ i = 0

The sum of x ^ i as x ranges over a finite field of cardinality q is equal to 0 if i < q - 1.

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theorem finite_field.frobenius_pow {K : Type u_1} [field K] [fintype K] {p : ℕ} [fact (nat.prime p)] [char_p K p] {n : ℕ} (hcard : fintype.card K = p ^ n) :

frobenius K p ^ n = 1

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theorem finite_field.expand_card {K : Type u_1} [field K] [fintype K] (f : polynomial K) :

⇑(polynomial.expand K (fintype.card K)) f = f ^ fintype.card K

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theorem zmod.sum_two_squares (p : ℕ) [hp : fact (nat.prime p)] (x : zmod p) :

∃ (a b : zmod p), a ^ 2 + b ^ 2 = x

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theorem char_p.sum_two_squares (R : Type u_1) [integral_domain R] (p : ℕ) [fact (0 < p)] [char_p R p] (x : ℤ) :

∃ (a b : ℕ), ↑a ^ 2 + ↑b ^ 2 = ↑x

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@[simp]

theorem zmod.pow_totient {n : ℕ} [fact (0 < n)] (x : units (zmod n)) :

x ^ n.totient = 1

The Fermat-Euler totient theorem. nat.modeq.pow_totient is an alternative statement of the same theorem.

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theorem nat.modeq.pow_totient {x n : ℕ} (h : x.coprime n) :

x ^ n.totient ≡ 1 [MOD n]

The Fermat-Euler totient theorem. zmod.pow_totient is an alternative statement of the same theorem.

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@[simp]

theorem zmod.pow_card {p : ℕ} [fact (nat.prime p)] (x : zmod p) :

x ^ p = x

A variation on Fermat's little theorem. See zmod.pow_card_sub_one_eq_one

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@[simp]

theorem zmod.frobenius_zmod (p : ℕ) [hp : fact (nat.prime p)] :

frobenius (zmod p) p = ring_hom.id (zmod p)

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@[simp]

theorem zmod.card_units (p : ℕ) [fact (nat.prime p)] :

fintype.card (units (zmod p)) = p - 1

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theorem zmod.units_pow_card_sub_one_eq_one (p : ℕ) [fact (nat.prime p)] (a : units (zmod p)) :

a ^ (p - 1) = 1

Fermat's Little Theorem: for every unit a of zmod p, we have a ^ (p - 1) = 1.

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theorem zmod.pow_card_sub_one_eq_one {p : ℕ} [fact (nat.prime p)] {a : zmod p} (ha : a ≠ 0) :

a ^ (p - 1) = 1

Fermat's Little Theorem: for all nonzero a : zmod p, we have a ^ (p - 1) = 1.

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theorem zmod.expand_card {p : ℕ} [fact (nat.prime p)] (f : polynomial (zmod p)) :

⇑(polynomial.expand (zmod p) p) f = f ^ p