Galois fields

If p is a prime number, and n a natural number, then GaloisField p n is defined as the splitting field of X^(p^n) - X over ZMod p. It is a finite field with p ^ n elements.

Main definition

• GaloisField p n is a field with p ^ n elements

Main Results

• GaloisField.algEquivGaloisField: Any finite field is isomorphic to some Galois field
• FiniteField.algEquivOfCardEq: Uniqueness of finite fields : algebra isomorphism
• FiniteField.ringEquivOfCardEq: Uniqueness of finite fields : ring isomorphism
• card_algHom_of_finrank_dvd: if [K:F] ∣ [L:F] then #(K →ₐ[F] L) = [K:F]
• nonempty_algHom_iff_finrank_dvd: (K →ₐ[F] L) is nonempty iff [K:F] ∣ [L:F]. This and the above result helps to classify the category of finite fields.

instance FiniteField.isSplittingField_sub (K : Type u_1) (F : Type u_2) [Field K] [Fintype K] [Field F] [Algebra F K] : Polynomial.IsSplittingField F K (Polynomial.X ^ Fintype.card K - Polynomial.X)

theorem galois_poly_separable {K : Type u_1} [CommRing K] (p q : ℕ) [CharP K p] (h : p ∣ q) : (Polynomial.X ^ q - Polynomial.X).Separable

def GaloisField (p : ℕ) [Fact (Nat.Prime p)] (n : ℕ) : Type

A finite field with p ^ n elements. Every field with the same cardinality is (non-canonically) isomorphic to this field.

Equations

• GaloisField p n = (Polynomial.X ^ p ^ n - Polynomial.X).SplittingField

@[instance_reducible]

noncomputable instance instInhabitedGaloisField (p : ℕ) [Fact (Nat.Prime p)] (n : ℕ) : Inhabited (GaloisField p n)

Equations

• instInhabitedGaloisField p n = { default := instInhabitedGaloisField._aux_1 p n }

@[instance_reducible]

noncomputable instance instFieldGaloisField (p : ℕ) [Fact (Nat.Prime p)] (n : ℕ) : Field (GaloisField p n)

Equations

• One or more equations did not get rendered due to their size.

instance instCharPGaloisField (p : ℕ) [Fact (Nat.Prime p)] (n : ℕ) : CharP (GaloisField p n) p

@[instance_reducible]

noncomputable instance instAlgebraZModGaloisField (p : ℕ) [Fact (Nat.Prime p)] (n : ℕ) : Algebra (ZMod p) (GaloisField p n)

Equations

• instAlgebraZModGaloisField p n = { smul := instAlgebraZModGaloisField._aux_1 p n, algebraMap := instAlgebraZModGaloisField._aux_3 p n, commutes' := ⋯, smul_def' := ⋯ }

instance instFiniteGaloisField (p : ℕ) [Fact (Nat.Prime p)] (n : ℕ) : Finite (GaloisField p n)

instance instFiniteDimensionalZModGaloisField (p : ℕ) [Fact (Nat.Prime p)] (n : ℕ) : FiniteDimensional (ZMod p) (GaloisField p n)

instance instIsSplittingFieldZModGaloisFieldHSubPolynomialHPowNatX (p : ℕ) [Fact (Nat.Prime p)] (n : ℕ) : Polynomial.IsSplittingField (ZMod p) (GaloisField p n) (Polynomial.X ^ p ^ n - Polynomial.X)

theorem GaloisField.finrank (p : ℕ) [h_prime : Fact (Nat.Prime p)] {n : ℕ} (h : n ≠ 0) : Module.finrank (ZMod p) (GaloisField p n) = n

theorem GaloisField.card (p : ℕ) [h_prime : Fact (Nat.Prime p)] (n : ℕ) (h : n ≠ 0) : Nat.card (GaloisField p n) = p ^ n

theorem GaloisField.splits_zmod_X_pow_sub_X (p : ℕ) [h_prime : Fact (Nat.Prime p)] : (Polynomial.X ^ p - Polynomial.X).Splits

noncomputable def GaloisField.equivZmodP (p : ℕ) [h_prime : Fact (Nat.Prime p)] : GaloisField p 1 ≃ₐ[ZMod p] ZMod p

A Galois field with exponent 1 is equivalent to ZMod

Equations

• GaloisField.equivZmodP p = (Polynomial.IsSplittingField.algEquiv (ZMod p) (Polynomial.X ^ p ^ 1 - Polynomial.X)).symm

theorem FiniteField.splits_X_pow_card_sub_X (p : ℕ) [h_prime : Fact (Nat.Prime p)] {K : Type u_1} [Field K] [Fintype K] [Algebra (ZMod p) K] : (Polynomial.map (algebraMap (ZMod p) K) (Polynomial.X ^ Fintype.card K - Polynomial.X)).Splits

theorem FiniteField.isSplittingField_of_card_eq (p : ℕ) [h_prime : Fact (Nat.Prime p)] (n : ℕ) {K : Type u_1} [Field K] [Fintype K] [Algebra (ZMod p) K] (h : Fintype.card K = p ^ n) : Polynomial.IsSplittingField (ZMod p) K (Polynomial.X ^ p ^ n - Polynomial.X)

noncomputable def GaloisField.algEquivGaloisFieldOfFintype (p : ℕ) [h_prime : Fact (Nat.Prime p)] (n : ℕ) {K : Type u_1} [Field K] [Fintype K] [Algebra (ZMod p) K] (h : Fintype.card K = p ^ n) : K ≃ₐ[ZMod p] GaloisField p n

Any finite field is (possibly noncanonically) isomorphic to some Galois field.

Equations

• GaloisField.algEquivGaloisFieldOfFintype p n h = Polynomial.IsSplittingField.algEquiv K (Polynomial.X ^ p ^ n - Polynomial.X)

theorem FiniteField.splits_X_pow_nat_card_sub_X (p : ℕ) [h_prime : Fact (Nat.Prime p)] {K : Type u_1} [Field K] [Algebra (ZMod p) K] [Finite K] : (Polynomial.map (algebraMap (ZMod p) K) (Polynomial.X ^ Nat.card K - Polynomial.X)).Splits

theorem FiniteField.isSplittingField_of_nat_card_eq (p : ℕ) [h_prime : Fact (Nat.Prime p)] (n : ℕ) {K : Type u_1} [Field K] [Algebra (ZMod p) K] (h : Nat.card K = p ^ n) : Polynomial.IsSplittingField (ZMod p) K (Polynomial.X ^ p ^ n - Polynomial.X)

theorem Polynomial.splits_X_pow_nat_card_sub_X {K : Type u_1} [Field K] : (X ^ Nat.card K - X).Splits

@[instance 100]

instance GaloisField.instIsGaloisOfFinite {K : Type u_2} {K' : Type u_3} [Field K] [Field K'] [Finite K'] [Algebra K K'] : IsGalois K K'

noncomputable def GaloisField.algEquivGaloisField (p : ℕ) [h_prime : Fact (Nat.Prime p)] (n : ℕ) {K : Type u_1} [Field K] [Algebra (ZMod p) K] (h : Nat.card K = p ^ n) : K ≃ₐ[ZMod p] GaloisField p n

Any finite field is (possibly noncanonically) isomorphic to some Galois field.

Equations

• GaloisField.algEquivGaloisField p n h = Polynomial.IsSplittingField.algEquiv K (Polynomial.X ^ p ^ n - Polynomial.X)

theorem FiniteField.algebraMap_norm_eq_pow {K : Type u_1} {K' : Type u_2} [Field K] [Field K'] [Algebra K K'] [Finite K'] {x : K'} : (algebraMap K K') ((Algebra.norm K) x) = x ^ ((Nat.card K' - 1) / (Nat.card K - 1))

theorem FiniteField.unitsMap_norm_surjective (K : Type u_1) (K' : Type u_2) [Field K] [Field K'] [Algebra K K'] [Finite K'] : Function.Surjective ⇑(Units.map (Algebra.norm K))

theorem FiniteField.norm_surjective (K : Type u_1) (K' : Type u_2) [Field K] [Field K'] [Algebra K K'] [Finite K'] : Function.Surjective ⇑(Algebra.norm K)

noncomputable def FiniteField.algEquivOfCardEq {K : Type u_1} {K' : Type u_2} [Field K] [Field K'] [Fintype K] [Fintype K'] (p : ℕ) [h_prime : Fact (Nat.Prime p)] [Algebra (ZMod p) K] [Algebra (ZMod p) K'] (hKK' : Fintype.card K = Fintype.card K') : K ≃ₐ[ZMod p] K'

Uniqueness of finite fields: Any two finite fields of the same cardinality are (possibly noncanonically) isomorphic

Equations

• One or more equations did not get rendered due to their size.

noncomputable def FiniteField.ringEquivOfCardEq {K : Type u_1} {K' : Type u_2} [Field K] [Field K'] [Fintype K] [Fintype K'] (hKK' : Fintype.card K = Fintype.card K') : K ≃+* K'

Uniqueness of finite fields: Any two finite fields of the same cardinality are (possibly noncanonically) isomorphic

Equations

• One or more equations did not get rendered due to their size.

theorem FiniteField.pow_finrank_eq_natCard (p : ℕ) [Fact (Nat.Prime p)] (k : Type u_3) [AddCommGroup k] [Finite k] [Module (ZMod p) k] : p ^ Module.finrank (ZMod p) k = Nat.card k

theorem FiniteField.pow_finrank_eq_card (p : ℕ) [Fact (Nat.Prime p)] (k : Type u_3) [AddCommGroup k] [Fintype k] [Module (ZMod p) k] : p ^ Module.finrank (ZMod p) k = Fintype.card k

theorem FiniteField.nonempty_algHom_of_finrank_dvd {F : Type u_3} {K : Type u_4} {L : Type u_5} [Field F] [Field K] [Algebra F K] [Field L] [Algebra F L] [Finite L] (h : Module.finrank F K ∣ Module.finrank F L) : Nonempty (K →ₐ[F] L)

theorem FiniteField.natCard_algHom_of_finrank_dvd {F : Type u_3} {K : Type u_4} {L : Type u_5} [Field F] [Field K] [Algebra F K] [Field L] [Algebra F L] [Finite L] (h : Module.finrank F K ∣ Module.finrank F L) : Nat.card (K →ₐ[F] L) = Module.finrank F K

theorem FiniteField.card_algHom_of_finrank_dvd {F : Type u_3} {K : Type u_4} {L : Type u_5} [Field F] [Field K] [Algebra F K] [Field L] [Algebra F L] [Finite L] [Finite K] (h : Module.finrank F K ∣ Module.finrank F L) : Fintype.card (K →ₐ[F] L) = Module.finrank F K

theorem FiniteField.nonempty_algHom_iff_finrank_dvd {F : Type u_3} {K : Type u_4} {L : Type u_5} [Field F] [Field K] [Algebra F K] [Field L] [Algebra F L] [Finite L] : Nonempty (K →ₐ[F] L) ↔ Module.finrank F K ∣ Module.finrank F L