Subfield Ontology

FLOW ALGEBRA
TITLE: Subfield Ontology
VERSION: 4.0
DOMAIN: Mathematics / Algebraic Entities / Field Theory
SOURCE_URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/algebraic-entities/subfield-ontology.html
AUTHORSHIP: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library
SOURCE_ALIGNMENT: Independently constructed; source-page content not verified.

1. PURPOSE
Model subfields as typed mathematical entities, specifying membership, closure, field embeddings, generated subfields, characteristic, extensions, and logical inference.

2. ENTITY TYPES
ENTITY Field: a commutative unital ring in which every nonzero element has a multiplicative inverse, with 0 != 1.
ENTITY Subfield: a subset K of a field F which is itself a field under operations inherited from F, with the same zero and unity.
ENTITY FieldElement: member of a field carrier set.
ENTITY AdditiveIdentity: 0_F.
ENTITY MultiplicativeIdentity: 1_F.
ENTITY AdditiveInverse: -a.
ENTITY MultiplicativeInverse: a^(-1), for a != 0.
ENTITY FieldEmbedding: injective unital field homomorphism.
ENTITY FieldExtension: pair K subseteq F of fields.
ENTITY PrimeSubfield: smallest subfield of a field.
ENTITY GeneratedSubfield: intersection of subfields containing a specified subset.
ENTITY IntermediateField: field E with K subseteq E subseteq F.
ENTITY Characteristic: additive order of 1, either zero or prime for fields.
ENTITY AlgebraicElement: element satisfying a nonzero polynomial over base field.
ENTITY TranscendentalElement: element satisfying no nonzero polynomial over base field.
ENTITY FieldAutomorphism: field isomorphism F -> F.
ENTITY FixedField: elements fixed by a specified group of automorphisms.

3. TYPED RELATIONSHIPS
RELATION is_subfield_of(K:Field,F:Field) -> Boolean.
RELATION is_subset_of(K:Set,F:Set) -> Boolean.
RELATION contains_identity(K:Field,one:FieldElement) -> Boolean.
RELATION closed_under_addition(K:Field) -> Boolean.
RELATION closed_under_multiplication(K:Field) -> Boolean.
RELATION closed_under_additive_inverse(K:Field) -> Boolean.
RELATION closed_under_nonzero_inverse(K:Field) -> Boolean.
RELATION embeds_into(K:Field,F:Field) -> Boolean.
RELATION extends(F:Field,K:Field) -> Boolean.
RELATION generated_by(K:Field,S:Set,F:Field) -> Boolean.
RELATION intermediate_between(E:Field,K:Field,F:Field) -> Boolean.
RELATION has_characteristic(F:Field,p:Integer) -> Boolean.
RELATION is_algebraic_over(a:FieldElement,K:Field) -> Boolean.
RELATION is_fixed_by(a:FieldElement,G:AutomorphismGroup) -> Boolean.

4. FORMAL SUBFIELD TEST
PRECONDITION: F is a field, K subseteq F.
SUBFIELD_TEST(K,F):
  (1) 1_F in K;
  (2) forall a,b in K, a-b in K;
  (3) forall a,b in K, a*b in K;
  (4) forall a in K with a != 0_F, a^(-1) in K.
CONCLUSION: K is a subfield of F if and only if all four conditions hold.
EQUIVALENT_TEST: K is nonempty, closed under inherited addition and multiplication, additive inverses and nonzero multiplicative inverses, and contains 1_F.
NOTE: Closure under division means a/b in K for a,b in K and b != 0.
NOTE: For fields, a unital homomorphism is injective because its kernel is a proper ideal.

5. STRUCTURAL AXIOMS
AXIOM SF01: is_subfield_of(K,F) implies is_field(K) and is_field(F).
AXIOM SF02: is_subfield_of(K,F) implies K subseteq F.
AXIOM SF03: is_subfield_of(K,F) implies 0_K = 0_F and 1_K = 1_F.
AXIOM SF04: is_subfield_of(K,F) implies inherited operations agree on K.
AXIOM SF05: is_subfield_of(F,F) for every field F.
AXIOM SF06: is_subfield_of(K,E) and is_subfield_of(E,F) imply is_subfield_of(K,F).
AXIOM SF07: is_subfield_of(K,F) and is_subfield_of(F,K) imply K=F, for concrete nested carriers with inherited operations.
AXIOM SF08: intersection of a nonempty family of subfields of F is a subfield of F.
AXIOM SF09: union of two subfields need not be a subfield.
AXIOM SF10: the union of a chain of subfields of F is a subfield of F.
AXIOM SF11: every field has a unique prime subfield.
AXIOM SF12: a field of characteristic zero has prime subfield isomorphic to Q.
AXIOM SF13: a field of characteristic p>0 has prime subfield isomorphic to F_p, where p is prime.
AXIOM SF14: subfields have the same characteristic as their ambient field.
AXIOM SF15: an injective unital field homomorphism identifies its domain with a subfield of its codomain.
AXIOM SF16: a field extension K subseteq F is equivalently a specified inclusion of fields.

6. INFERENCE RULES
RULE R01: field(F) and subfield_test(K,F) => subfield(K,F).
RULE R02: subfield(K,F) => subset(K,F).
RULE R03: subfield(K,F) => field(K).
RULE R04: subfield(K,F) => contains(K,0_F).
RULE R05: subfield(K,F) => contains(K,1_F).
RULE R06: subfield(K,F) and a,b in K => a+b in K.
RULE R07: subfield(K,F) and a,b in K => a*b in K.
RULE R08: subfield(K,F) and a in K => -a in K.
RULE R09: subfield(K,F) and a in K and a != 0 => inverse(a) in K.
RULE R10: subfield(K,F) and a,b in K and b != 0 => a/b in K.
RULE R11: subfield(K,E) and subfield(E,F) => subfield(K,F).
RULE R12: subfield(K,F) and subfield(E,F) => subfield(K intersect E,F).
RULE R13: family_of_subfields(C,F) and C nonempty => subfield(intersection(C),F).
RULE R14: chain_of_subfields(C,F) and C nonempty => subfield(union(C),F).
RULE R15: subfield(K,F) => characteristic(K)=characteristic(F).
RULE R16: field(F) => subfield(prime_subfield(F),F).
RULE R17: characteristic(F)=0 => prime_subfield(F) isomorphic_to Q.
RULE R18: characteristic(F)=p and p>0 => prime(p) and prime_subfield(F) isomorphic_to F_p.
RULE R19: subfield(K,F) and subset(S,F) => subfield(generated_field(K,S,F),F).
RULE R20: subfield(K,F) and subset(S,F) => subset(S,generated_field(K,S,F)).
RULE R21: subfield(K,F) and subfield(E,F) and K subseteq E and S subseteq E => generated_field(K,S,F) subseteq E.
RULE R22: subfield(K,F) and a in F => K(a) is the smallest subfield of F containing K and a.
RULE R23: subfield(K,F) and a algebraic_over K => [K(a):K] finite.
RULE R24: subfield(K,F) and a transcendental_over K => K(a) isomorphic_to K(t), the rational function field.
RULE R25: field_embedding(phi,K,F) => image(phi) is a subfield of F.
RULE R26: subfield(K,F) => F is a vector space over K.
RULE R27: finite_extension(F,K) => [F:K] is a positive integer.
RULE R28: subfield(K,E) and subfield(E,F) and finite_extension(F,E) and finite_extension(E,K) => [F:K]=[F:E]*[E:K].
RULE R29: subfield(K,F) and subfield(E,F) => compositum(K,E,F) is a subfield of F.
RULE R30: subgroup(G,Aut(F)) => fixed_field(F,G) is a subfield of F.
RULE R31: subfield(K,F) and finite(F) => finite(K) and |K|=p^d for some prime p and positive integer d.
RULE R32: finite_field(F) with |F|=p^n => for each d dividing n there is exactly one subfield of F with p^d elements.

7. VALIDATION CONSTRAINTS
CHECK V01: F must satisfy all field axioms, including 0 != 1.
CHECK V02: K must be a subset of F for literal subfield claims.
CHECK V03: K must contain the ambient multiplicative identity 1_F.
CHECK V04: K must be closed under subtraction.
CHECK V05: K must be closed under multiplication.
CHECK V06: K must contain inverses of all its nonzero elements.
CHECK V07: Do not classify a non-unital subring as a subfield.
CHECK V08: Distinguish a subfield inclusion from an abstract field embedding.
CHECK V09: An arbitrary union of subfields is not necessarily a subfield.
CHECK V10: A subfield cannot have a different characteristic from its parent field.
CHECK V11: Finite field subfield orders must be prime powers with exponent dividing the parent exponent.
CHECK V12: Finite extension degrees require finite-dimensionality.
CHECK V13: Algebraic extensions need not have finite total degree.
CHECK V14: A field cannot be an ordered field if its characteristic is positive.
CHECK V15: Any claim about a unique subfield of order p^d presumes an ambient finite field of order p^n with d|n.

8. MATHEMATICAL EXAMPLES
EXAMPLE E01: Q is a subfield of R.
EXAMPLE E02: R is a subfield of C.
EXAMPLE E03: Q is a subfield of C by transitivity.
EXAMPLE E04: Z is not a subfield of Q because 2 has no inverse in Z.
EXAMPLE E05: Q(sqrt(2)) is a subfield of R.
EXAMPLE E06: Q(i) is a subfield of C.
EXAMPLE E07: F_p is the prime subfield of every field of characteristic p.
EXAMPLE E08: F_4 is a subfield of F_16, as 2 divides 4.
EXAMPLE E09: F_8 is not a subfield of F_16, as 3 does not divide 4.
EXAMPLE E10: F_9 is a subfield of F_81, as 2 divides 4.
EXAMPLE E11: Q(t) is the field of rational functions in an indeterminate t over Q.
EXAMPLE E12: The real algebraic numbers form a subfield of R.
EXAMPLE E13: The algebraic numbers form a subfield of C.
EXAMPLE E14: For complex conjugation, the fixed field within C is R.
EXAMPLE E15: The intersection Q(sqrt(2)) intersect Q(sqrt(3)) inside R is Q.
EXAMPLE E16: Q(sqrt(2),sqrt(3)) is the compositum of Q(sqrt(2)) and Q(sqrt(3)) inside R.

9. SEMANTIC EQUIVALENCE
SEMANTIC_EQUIVALENCE: subfield <=> included field with inherited operations and common identity.
SEMANTIC_EQUIVALENCE: field extension F/K <=> specified inclusion K subseteq F of fields.
SEMANTIC_EQUIVALENCE: subfield generated by S over K <=> smallest intermediate field containing S.
SEMANTIC_EQUIVALENCE: prime subfield <=> field generated by 1.
SEMANTIC_DISTINCTION: subring != subfield in general.
SEMANTIC_DISTINCTION: field embedding != literal set inclusion, unless an identification is specified.
SEMANTIC_DISTINCTION: intermediate field != necessarily normal or Galois extension.
SEMANTIC_DISTINCTION: algebraic extension != necessarily finite extension.

10. KNOWLEDGE GRAPH TRIPLES
TRIPLE (Subfield, is_a, Field).
TRIPLE (Subfield, is_a, AlgebraicStructure).
TRIPLE (Subfield, is_related_to, FieldExtension).
TRIPLE (Subfield, is_related_to, FieldEmbedding).
TRIPLE (Subfield, has_property, ClosureUnderAddition).
TRIPLE (Subfield, has_property, ClosureUnderMultiplication).
TRIPLE (Subfield, has_property, ClosureUnderAdditiveInverse).
TRIPLE (Subfield, has_property, ClosureUnderNonzeroInverse).
TRIPLE (Subfield, shares_identity_with, AmbientField).
TRIPLE (PrimeSubfield, is_a, Subfield).
TRIPLE (GeneratedSubfield, is_a, Subfield).
TRIPLE (IntermediateField, is_a, Subfield).
TRIPLE (Subfield, inherits_characteristic_from, AmbientField).
TRIPLE (FieldExtension, supports, VectorSpaceStructure).

11. FLOW EXECUTION
INPUT: ambient field F, candidate subset K, operations, proof evidence, optional generating set S.
STEP 1: Verify F is a field and K subseteq F.
STEP 2: Verify 1_F in K.
STEP 3: Check subtraction closure on K.
STEP 4: Check multiplication closure on K.
STEP 5: Check inverse closure for every nonzero element of K.
STEP 6: If all checks pass, emit is_subfield_of(K,F)=true; otherwise emit false or unknown according to evidence.
STEP 7: Derive inherited identities, characteristic and extension relationships.
STEP 8: If S supplied, construct or describe generated subfield and its minimality.
STEP 9: Validate inferred relations against constraints.
STEP 10: Record proof provenance, confidence and update metadata.
OUTPUT: typed subfield assessment, proof obligations, derived relations, and provenance.

12. CONFIDENCE
CONFIDENCE: formal theorem with verified hypotheses -> 1.00.
CONFIDENCE: machine-checked algebraic certificate -> 1.00 subject to trusted checker assumptions.
CONFIDENCE: incomplete or unverified premise -> UNKNOWN; do not invent numerical certainty.
CONFIDENCE: illustrative examples are not substitutes for general proof.

13. FLOW_HASH
FLOW_HASH: compute SHA-256 over canonical UTF-8 serialization of the flow, excluding the hash field itself.
FLOW_HASH_STATUS: NOT_COMPUTED.
FLOW_HASH_POLICY: recompute after any material edit; do not fabricate a digest.

14. UPDATE_MODEL
UPDATE_MODEL.version: 4.0.
UPDATE_MODEL.strategy: versioned immutable publication with explicit revisions.
UPDATE_MODEL.trigger: correction, added theorem, changed entity or relationship, improved proof, updated source page.
UPDATE_MODEL.validation: rerun axiom consistency, typing, example and rule checks.
UPDATE_MODEL.provenance: preserve source URL, author credit, date, and change history.

15. AI_CONSUMPTION
AI_CONSUMPTION.format: plain_text/flow_algebra.
AI_CONSUMPTION.encoding: UTF-8.
AI_CONSUMPTION.language: en-GB.
AI_CONSUMPTION.entity_resolution: distinguish field, subfield, subset, subring, embedding, extension.
AI_CONSUMPTION.reasoning: apply inference rules only when prerequisites are verified.
AI_CONSUMPTION.uncertainty: preserve unknown and unsupported claims.
AI_CONSUMPTION.explainability: attach axiom/rule IDs and mathematical witnesses.
AI_CONSUMPTION.interoperability: export typed entities and triples to semantic graph systems.
AI_CONSUMPTION.citation: cite this flow and original source page distinctly.

END FLOW ALGEBRA