Documentation

Mathlib.FieldTheory.Finite.GaloisField

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 #

Main Results #

theorem galois_poly_separable {K : Type u_1} [Field K] (p q : ) [CharP K p] (h : p q) :
def GaloisField (p : ) [Fact (Nat.Prime p)] (n : ) :

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

Equations
Instances For
    instance GaloisField.instCharP (p : ) [h_prime : Fact (Nat.Prime p)] (n : ) :
    instance GaloisField.instFinite (p : ) [h_prime : Fact (Nat.Prime p)] (n : ) :
    theorem GaloisField.finrank (p : ) [h_prime : Fact (Nat.Prime p)] {n : } (h : n 0) :
    theorem GaloisField.card (p : ) [h_prime : Fact (Nat.Prime p)] (n : ) (h : n 0) :

    A Galois field with exponent 1 is equivalent to ZMod

    Equations
    Instances For
      @[deprecated FiniteField.splits_X_pow_card_sub_X (since := "2024-11-12")]

      Alias of FiniteField.splits_X_pow_card_sub_X.

      @[deprecated FiniteField.isSplittingField_of_card_eq (since := "2024-11-12")]
      theorem GaloisField.isSplittingField_of_card_eq (p : ) [h_prime : Fact (Nat.Prime p)] (n : ) {K : Type u_1} [Field K] [Fintype K] [Algebra (ZMod p) K] (h : Fintype.card K = p ^ n) :

      Alias of FiniteField.isSplittingField_of_card_eq.

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

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

      Equations
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        @[instance 100]
        instance GaloisField.instIsGaloisOfFinite {K : Type u_2} {K' : Type u_3} [Field K] [Field K'] [Finite K'] [Algebra K K'] :
        def GaloisField.algEquivGaloisField (p : ) [h_prime : Fact (Nat.Prime p)] (n : ) {K : Type u_1} [Field K] [Algebra (ZMod p) K] (h : Nat.card K = p ^ n) :

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

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

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

          Equations
          • One or more equations did not get rendered due to their size.
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            def FiniteField.ringEquivOfCardEq {K : Type u_1} [Field K] [Fintype K] {K' : Type u_2} [Field 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 non canonically) isomorphic

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