Field Ontology

FLOW Field_Ontology_v1.0

PAGE:
  title: "Field Ontology"
  canonical_url: "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/field-ontology.html"
  language: "en-GB"
  domain: "Mathematics > Algebra > Abstract Algebra > Fields"

DEFINE FieldOntology := {

  CORE_IDENTITY: {
    entity_type: "Mathematical Ontology",
    primary_entity: Field,
    parent_entities: [Ring, AlgebraicStructure, SetWithOperations],
    related_entities: [
      Set,
      BinaryOperation,
      Addition,
      Multiplication,
      AdditiveIdentity,
      MultiplicativeIdentity,
      AdditiveInverse,
      MultiplicativeInverse,
      Commutativity,
      Associativity,
      Distributivity,
      IntegralDomain,
      DivisionRing,
      Subfield,
      FieldExtension,
      FiniteField,
      OrderedField
    ]
  },

  DEFINITION: {
    meaning:
      "A field is a set equipped with addition and multiplication in which the usual arithmetic laws hold and every nonzero element has a multiplicative inverse.",
    structure:
      (F,+,*),
    requirements: [
      (F,+) is an abelian group,
      (F\{0},*) is an abelian group,
      multiplication_distributes_over_addition
    ]
  },

  SEMANTIC_PRIMITIVES: {
    CarrierSet:
      F,
    Addition:
      binary_operation_on_F,
    Multiplication:
      binary_operation_on_F,
    Zero:
      additive_identity,
    One:
      multiplicative_identity,
    AdditiveInverse:
      -a,
    MultiplicativeInverse:
      a^(-1)_for_a!=0
  },

  CLOSURE_AXIOMS: {
    addition:
      FOR_ALL a,b IN F:
        a+b IN F,
    multiplication:
      FOR_ALL a,b IN F:
        a*b IN F
  },

  ADDITIVE_GROUP_AXIOMS: {
    associativity:
      (a+b)+c=a+(b+c),

    identity:
      EXISTS 0 IN F SUCH_THAT a+0=a,

    inverses:
      FOR_ALL a IN F:
        EXISTS -a IN F SUCH_THAT a+(-a)=0,

    commutativity:
      a+b=b+a
  },

  MULTIPLICATIVE_GROUP_AXIOMS: {
    domain:
      F\{0},

    associativity:
      (a*b)*c=a*(b*c),

    identity:
      EXISTS 1 IN F WITH 1!=0
      SUCH_THAT a*1=a,

    inverses:
      FOR_ALL a!=0:
        EXISTS a^(-1)
        SUCH_THAT a*a^(-1)=1,

    commutativity:
      a*b=b*a
  },

  DISTRIBUTIVE_AXIOMS: {
    left_distributive:
      a*(b+c)=a*b+a*c,

    right_distributive:
      (a+b)*c=a*c+b*c
  },

  DISTINCT_IDENTITIES: {
    field_requirement:
      0!=1
  },

  DERIVED_SUBTRACTION: {
    a-b:
      a+(-b)
  },

  DERIVED_DIVISION: {
    a/b:
      a*b^(-1),
    condition:
      b!=0
  },

  ZERO_PRODUCT_PROPERTY: {
    IF a*b=0:
      a=0 OR b=0,

    consequence:
      every_field_is_an_integral_domain
  },

  CANCELLATION_LAWS: {
    additive:
      a+c=b+c
      -> a=b,

    multiplicative:
      a*c=b*c AND c!=0
      -> a=b
  },

  CLASSIFICATION: {

    PrimeFieldCharacteristicZero: {
      isomorphic_to:
        RationalNumbers
    },

    PrimeFieldCharacteristicP: {
      isomorphic_to:
        FiniteField_F_p
      WHERE p_is_prime
    },

    FiniteField: {
      finite_number_of_elements:
        TRUE
    },

    InfiniteField: {
      finite_number_of_elements:
        FALSE
    },

    OrderedField: {
      compatible_total_order:
        TRUE
    },

    AlgebraicallyClosedField: {
      every_nonconstant_polynomial_has_root_in_field
    },

    RealClosedField: {
      ordered_field_with_specific_algebraic_closure_properties
    }
  },

  EXAMPLES: {
    RationalNumbers:
      field=TRUE,

    RealNumbers:
      field=TRUE,

    ComplexNumbers:
      field=TRUE,

    Integers:
      field=FALSE
      because_nonzero_elements_need_not_have_multiplicative_inverses_in_Integers,

    IntegersModuloPrimeP:
      field=TRUE,

    IntegersModuloCompositeN:
      generally_field=FALSE
  },

  NONEXAMPLES: {
    NaturalNumbers:
      lacks_additive_inverses,

    Integers:
      lacks_general_multiplicative_inverses,

    Matrices_nxn:
      generally_not_field_due_to_noncommutativity_and_noninvertible_nonzero_elements,

    Quaternions:
      division_ring_but_not_field_due_to_noncommutativity
  },

  SUBFIELD: {
    definition:
      K SUBSET_OR_EQUAL F,

    condition:
      K_is_field_under_operations_inherited_from_F,

    subfield_test:
      nonempty_K
      AND closed_under_subtraction
      AND closed_under_division_by_nonzero_elements
  },

  PRIME_SUBFIELD: {
    definition:
      smallest_subfield_contained_in_field,

    characteristic_zero:
      isomorphic_to_Rationals,

    characteristic_p:
      isomorphic_to_F_p
  },

  CHARACTERISTIC: {
    definition:
      least_positive_integer_n
      SUCH_THAT n*1=0,

    IF no_positive_integer_exists:
      characteristic=0,

    theorem:
      characteristic_of_field_is_0_or_prime
  },

  FINITE_FIELD: {
    cardinality:
      p^n
      for_prime_p_and_integer_n>=1,

    uniqueness:
      up_to_isomorphism_there_is_unique_finite_field_of_order_p^n
  },

  FIELD_EXTENSION: {
    notation:
      E/F,

    meaning:
      F_is_subfield_of_E,

    extension_degree:
      [E:F]
      as_vector_space_dimension_of_E_over_F
  },

  SIMPLE_EXTENSION: {
    form:
      F(alpha),

    meaning:
      smallest_field_containing_F_and_element_alpha
  },

  ALGEBRAIC_ELEMENT: {
    alpha_over_F:
      satisfies_nonzero_polynomial_in_F[x]
  },

  TRANSCENDENTAL_ELEMENT: {
    alpha_over_F:
      satisfies_no_nonzero_polynomial_in_F[x]
  },

  ALGEBRAIC_EXTENSION: {
    every_element_of_E_is_algebraic_over_F
  },

  FINITE_EXTENSION: {
    [E:F] finite
      -> algebraic_extension
  },

  MINIMAL_POLYNOMIAL: {
    for_algebraic_alpha_over_F:
      unique_monic_irreducible_polynomial_m_alpha(x)
      in_F[x]
      with_m_alpha(alpha)=0
  },

  IRREDUCIBLE_POLYNOMIAL_RELATION: {
    polynomial:
      irreducible_over_F
      IF nonconstant_and_not_factorable_into_lower_degree_nonunits_in_F[x]
  },

  QUOTIENT_FIELD_CONSTRUCTION: {
    IF p(x)_irreducible_in_F[x]:
      F[x]/(p(x))
      is_field
  },

  FIELD_OF_FRACTIONS: {
    integral_domain_D:
      embeds_into_smallest_field_Frac(D),

    example:
      Frac(Integers)=RationalNumbers
  },

  POLYNOMIAL_RING_RELATION: {
    F_field:
      F[x] is_EuclideanDomain,
      principal_ideal_domain,
      unique_factorisation_domain
  },

  RATIONAL_FUNCTION_FIELD: {
    F(x):
      field_of_fractions_of_F[x],

    elements:
      ratios_of_polynomials_with_nonzero_denominator
  },

  ORDERED_FIELD: {
    total_order:
      <,

    compatibility: [
      a<b -> a+c<b+c,
      0<a AND 0<b -> 0<a*b
    ]
  },

  ORDERED_FIELD_CONSEQUENCES: {
    characteristic:
      0,

    square_nonnegative:
      a^2>=0
  },

  ALGEBRAICALLY_CLOSED_FIELD: {
    definition:
      every_nonconstant_polynomial_in_F[x]_has_root_in_F,

    equivalent:
      every_nonconstant_polynomial_splits_into_linear_factors
  },

  ALGEBRAIC_CLOSURE: {
    of_field_F:
      algebraically_closed_extension_overline_F
      algebraic_over_F
  },

  COMPLEX_FIELD_RELATION: {
    ComplexNumbers:
      algebraically_closed,

    RealNumbers:
      not_algebraically_closed
  },

  REAL_CLOSED_FIELD_RELATION: {
    RealNumbers:
      real_closed,

    property:
      adjoining_i_produces_algebraic_closure
  },

  HOMOMORPHISM: {
    field_homomorphism_phi:F->K:

      phi(a+b)=phi(a)+phi(b),

      phi(a*b)=phi(a)*phi(b),

      phi(1_F)=1_K
  },

  FIELD_HOMOMORPHISM_PROPERTIES: {
    kernel:
      {0}
      for_nonzero_field_homomorphism,

    injective:
      TRUE_for_unital_field_homomorphisms
  },

  FIELD_ISOMORPHISM: {
    definition:
      bijective_field_homomorphism,

    consequence:
      preserves_all_field_structure
  },

  AUTOMORPHISM: {
    definition:
      field_isomorphism_from_F_to_itself
  },

  GALOIS_RELATION: {
    GaloisGroup(E/F):
      automorphisms_of_E_fixing_F,

    semantic_link:
      field_extensions_and_symmetry
  },

  VECTOR_SPACE_RELATION: {
    any_field_extension_E/F:
      E_is_vector_space_over_F
  },

  LINEAR_ALGEBRA_FOUNDATION: {
    vector_spaces:
      defined_over_fields,

    matrices:
      entries_often_taken_from_field,

    determinants:
      field_operations_support_standard_theory
  },

  EQUATION_SOLVING_RELATION: {
    linear_equation:
      a*x=b,

    IF a!=0:
      x=a^(-1)*b,

    field_property:
      guarantees_division_by_nonzero_coefficients
  },

  POLYNOMIAL_ROOT_RELATION: {
    root_existence:
      depends_on_field,

    example:
      x^2+1=0
      has_no_root_in_RealNumbers
      but_has_roots_in_ComplexNumbers
  },

  EXTENSION_REASONING: {
    purpose:
      enlarge_field_to_include_solutions_or_new algebraic_elements
  },

  MODULAR_FIELD: {
    F_p:
      integers_modulo_prime_p,

    operations:
      addition_and_multiplication_mod_p
  },

  FINITE_FIELD_MULTIPLICATIVE_GROUP: {
    F_q^*:
      cyclic_group_of_order_q-1
  },

  FROBENIUS_MAP: {
    finite_field_characteristic_p:
      phi(x)=x^p,

    property:
      field_endomorphism,

    finite_field:
      automorphism
  },

  SEMANTIC_INVARIANTS: {
    closure:
      required,

    zero_one_distinct:
      required,

    nonzero_inverse:
      required,

    multiplication_commutative:
      required
  },

  SYMBOLIC_RECOGNITION: {
    ACCEPT: [
      "Q with ordinary addition and multiplication",
      "R",
      "C",
      "F_5",
      "Z/7Z"
    ],

    REJECT_AS_FIELD: [
      "Z",
      "N",
      "Z/6Z",
      "quaternions under standard multiplication"
    ]
  },

  NATURAL_LANGUAGE_MAPPING: {
    "field":
      commutative_division_algebra_with_two_compatible_operations,

    "subfield":
      subset_that_is_itself_field_under_inherited_operations,

    "characteristic":
      additive_order_of_multiplicative_identity,

    "field extension":
      larger_field_containing_smaller_field
  },

  ERROR_DETECTION: {
    MissingInverse:
      nonzero_element_without_multiplicative_inverse
      -> reject_field_classification,

    NoncommutativeMultiplication:
      -> classify_as_division_ring_if_other_axioms_hold,

    ZeroEqualsOne:
      -> reject_nontrivial_field,

    CompositeModulusError:
      Z/nZ_called_field_for_composite_n
      -> flag,

    CharacteristicError:
      nonzero_nonprime_characteristic_claimed_for_field
      -> flag
  },

  ONTOLOGY_RELATIONS: {
    IS_A: [
      "Field IS_A CommutativeRing",
      "Field IS_A IntegralDomain",
      "FiniteField IS_A Field",
      "OrderedField IS_A Field"
    ],

    RELATED_TO: [
      "Field RELATED_TO VectorSpace",
      "Field RELATED_TO PolynomialRing",
      "Field RELATED_TO FieldExtension",
      "Field RELATED_TO GaloisTheory",
      "Field RELATED_TO LinearAlgebra",
      "Field RELATED_TO NumberTheory"
    ],

    DEPENDS_ON: [
      Set,
      BinaryOperations,
      GroupAxioms,
      RingAxioms
    ]
  },

  SEMANTIC_EQUIVALENCE: {
    field_isomorphism:
      equivalence_of_field_structure,

    examples:
      finite_fields_of_same_order_are_isomorphic,

    caution:
      identical_cardinality_does_not_imply_isomorphism_for_arbitrary_infinite_fields
  },

  CONFIDENCE: {
    classification_confidence:
      IF all_field_axioms_verified
      THEN 1.0,

    finite_field_confidence:
      IF cardinality_is_prime_power
      AND operations_verified
      THEN 1.0,

    extension_confidence:
      IF subfield_embedding_and_operations_verified
      THEN 1.0
  },

  AI_CONSUMPTION: {
    tasks: [
      recognise_field_candidate,
      verify_closure,
      verify_additive_abelian_group,
      verify_nonzero_multiplicative_abelian_group,
      verify_distributivity,
      identify_characteristic,
      detect_subfields,
      classify_finite_or_infinite,
      classify_ordered_or_algebraically_closed,
      analyse_field_extensions,
      construct_field_of_fractions,
      analyse_polynomial_roots_over_field,
      detect_field_axiom_failures,
      link_related_ontologies
    ],

    preferred_representation:
      typed_algebraic_structure_with_axiom_graph,

    reasoning_mode:
      recursive_axiom_verification_and_extension_analysis
  },

  RECURSIVE_REASONING: {

    FUNCTION AnalyseFieldCandidate(S,Plus,Times,Context):

      carrier := ParseSet(S)
      add := ParseOperation(Plus)
      mult := ParseOperation(Times)

      IF not VerifyClosure(carrier,add):
        RETURN NotField("addition not closed")

      IF not VerifyAbelianGroup(carrier,add):
        RETURN NotField("additive structure invalid")

      nonzero := carrier\{0}

      IF not VerifyAbelianGroup(nonzero,mult):
        RETURN NotField("nonzero multiplicative structure invalid")

      IF not VerifyDistributivity(carrier,add,mult):
        RETURN NotField("distributivity invalid")

      IF 0=1:
        RETURN NotField("trivial structure excluded")

      characteristic := DetermineCharacteristic(carrier,add,mult)

      properties := {
        characteristic:characteristic,
        finite:IsFinite(carrier),
        ordered:DetectCompatibleOrder(carrier),
        algebraically_closed:TestAlgebraicClosureWhenFeasible(carrier)
      }

      RETURN {
        entity:Field,
        carrier,
        operations:[add,mult],
        properties,
        verified:TRUE
      }
  },

  RECURSIVE_COHERENCE: {
    invariant_1:
      additive_structure_is_abelian_group=TRUE,

    invariant_2:
      nonzero_multiplicative_structure_is_abelian_group=TRUE,

    invariant_3:
      multiplication_distributes_over_addition=TRUE,

    invariant_4:
      zero_and_one_are_distinct=TRUE,

    invariant_5:
      characteristic_is_zero_or_prime=TRUE
  },

  FLOW_HASH: {
    semantic_signature_inputs: [
      ontology_name,
      field_definition,
      additive_axioms,
      multiplicative_axioms,
      distributive_axioms,
      characteristic_model,
      extension_relations
    ],

    purpose:
      "Detect semantic drift between revisions of the ontology."
  },

  UPDATE_MODEL: {
    version: "1.0",

    immutable_constraints: [
      additive_abelian_group,
      nonzero_multiplicative_abelian_group,
      distributivity,
      zero_not_equal_one,
      multiplicative_inverses_for_all_nonzero_elements
    ],

    revision_triggers: [
      semantic_conflict,
      RingOntology_update,
      GroupOntology_update,
      FieldExtensionOntology_update,
      representation_standard_change
    ]
  },

  KNOWLEDGE_GRAPH: {
    nodes: [
      Field,
      Ring,
      IntegralDomain,
      CarrierSet,
      Addition,
      Multiplication,
      Zero,
      One,
      Inverse,
      Subfield,
      FieldExtension,
      Characteristic,
      FiniteField
    ],

    edges: [
      Field -> IS_A -> Ring,
      Field -> IS_A -> IntegralDomain,
      Field -> HAS_CARRIER -> CarrierSet,
      Field -> HAS_OPERATION -> Addition,
      Field -> HAS_OPERATION -> Multiplication,
      Addition -> HAS_IDENTITY -> Zero,
      Multiplication -> HAS_IDENTITY -> One,
      NonzeroElement -> HAS_MULTIPLICATIVE_INVERSE -> Inverse,
      Subfield -> SUBSET_OF -> Field,
      FieldExtension -> CONTAINS -> Subfield,
      Field -> HAS_CHARACTERISTIC -> Characteristic,
      FiniteField -> IS_A -> Field
    ]
  },

  CROSS_ONTOLOGY_LINKS: {
    RingOntology:
      direct_parent,

    GroupOntology:
      additive_and_multiplicative_group_structures,

    IntegralDomainOntology:
      inherited_zero_product_properties,

    VectorSpaceOntology:
      scalar_field_foundation,

    PolynomialOntology:
      polynomial_rings_and_roots,

    FieldExtensionOntology:
      extension_structure,

    GaloisTheoryOntology:
      automorphism_and_extension_symmetry,

    NumberTheoryOntology:
      finite_fields_and_modular_arithmetic
  },

  TERMINATION_CONDITION: {
    ontology_complete_when: [
      field_defined,
      carrier_and_operations_defined,
      additive_axioms_defined,
      multiplicative_axioms_defined,
      distributivity_defined,
      characteristic_defined,
      subfield_and_extension_relations_defined,
      finite_and_ordered_field_classes_defined,
      error_detection_defined,
      cross_ontology_links_defined,
      recursive_coherence_verified
    ]
  }

}

EXECUTE Field_Ontology_v1.0:

  INPUT:
    mathematical_entity_or_query

  PIPELINE:
    Parse
    -> IdentifyCarrierSetAndOperations
    -> VerifyAdditionClosure
    -> VerifyAdditiveAbelianGroup
    -> VerifyMultiplicationClosureOnNonzeroElements
    -> VerifyMultiplicativeAbelianGroup
    -> VerifyDistributivity
    -> VerifyZeroAndOneDistinct
    -> DetermineCharacteristic
    -> ClassifyFiniteOrderedOrAlgebraicallyClosed
    -> AnalyseSubfieldsAndExtensions
    -> VerifySemanticConsistency
    -> LinkRelatedOntologies
    -> AssignConfidence
    -> EmitSemanticResult

  OUTPUT: {
    entity,
    classification,
    carrier_set,
    addition_operation,
    multiplication_operation,
    additive_identity,
    multiplicative_identity,
    inverse_structure,
    characteristic,
    finite_status,
    order_status,
    algebraic_closure_status,
    subfields,
    extensions,
    verification,
    ontology_relations,
    confidence,
    semantic_equivalence,
    flow_hash
  }

END FLOW Field_Ontology_v1.0