| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Studies algebraic structures where addition, subtraction, multiplication, and division (except by 0) are always possible and satisfy the field axioms. Includes extensions of fields, algebraic and transcendental elements, splitting fields, Galois groups, valuations, and topological/ordered field structures. Excludes rings without multiplicative inverses (except as substructures) and non-associative algebraic systems. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates over algebraic, arithmetic, and geometric scales: finite fields, number fields, function fields, local/global field behavior, degrees of extension, tower constructions, and infinite algebraic closures. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Field elements, additive and multiplicative identities, inverses, field extensions, bases, algebraic elements, transcendental elements, embeddings, automorphisms, valuations, minimal polynomials, splitting fields. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Commutativity of addition and multiplication, distributivity, existence of inverses, characteristic, extension degree, algebraicity/transcendence, normality, separability, completeness (for valued fields), residue fields. |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | Finite fields, number fields, function fields, algebraic extensions, transcendental extensions, separable and inseparable extensions, Galois extensions, local fields (e.g., ℚₚ), global fields (ℚ, function fields of curves), algebraically closed fields. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Current base field, chosen extension, degree of extension, minimal polynomial, automorphism group element, valuation value, residue characteristic, chosen embedding, coordinate representation under basis. |
| | Parameterization | How variables encode and represent the system’s state. | Encoded by polynomial generators, bases of extensions, minimal polynomials, valuation parameters, embedding maps, tower constructions, discriminants, norms and traces. |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Working only with separable extensions; assuming perfect fields; restricting to characteristic 0; ignoring ramification; reducing extension towers to simple extensions; focusing on finite extensions; treating infinite Galois groups via inverse limits. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Break down in inseparable or wildly ramified extensions; in positive characteristic pathologies; in infinite algebraic closures requiring cardinality considerations; when ignoring topological or valuation-theoretic structure for local fields. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Field axioms always hold; polynomial equations determine algebraic structure; extension degree is multiplicative in towers; homomorphisms preserve field operations; splitting fields exist; Galois groups capture full symmetry of algebraic equations. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes associativity and commutativity of multiplication; assumes existence of algebraic closures; assumes ability to construct bases for extensions; assumes polynomials encode extension data; assumes separability is generic unless field characteristic intervenes. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Field operations must not contradict axioms; extension structures must respect tower laws; automorphism groups must act compatibly; splitting fields must contain required roots; Galois correspondences must pair correctly with subgroup structures. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires harmony among polynomial theory, extension theory, Galois theory, valuation theory, number-field/function-field structures, and categorical formulations (e.g., adjunctions, functorial constructions). |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Polynomial factorization patterns; behavior of roots under field extensions; splitting-field formation; separability/inseparability behavior; degree of extensions; automorphism structure; ramification behavior in valued fields; norm/trace computations; residue-field interactions. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Limited by inability to factor arbitrary polynomials efficiently; difficulty detecting inseparability in large characteristic; computational hardness of Galois group determination; limits in numerically approximating roots; difficulty observing infinite extensions directly. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Extension degree; characteristic; valuation magnitude; ramification index; discriminant size; residue degree; order of automorphism groups; polynomial degree; norm and trace values; embedding count. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Polynomial factorization algorithms; field arithmetic engines; Galois group solvers; valuation and completion tools; number-field computation systems (PARI/GP, Magma, Sage); discriminant calculators; norm/trace computation tools; root-approximation solvers. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Algebraic element defined as root of a polynomial; separability defined via derivative nonvanishing; extension degree defined by basis size; Galois extension defined as normal + separable; valuation defined as ordered-group mapping; discriminant defined via determinant of trace-forms. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Factoring polynomials; computing minimal polynomials; building extension towers; computing splitting fields; calculating automorphism groups; evaluating norms and traces; computing valuations; performing completions; determining ramification. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Standardized polynomial factorization workflows; canonical extension-building procedures; structured extraction of Galois groups; fixed protocols for computing discriminants; controlled valuation sampling; consistent norm/trace calculations; uniform tower construction. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Sampling random polynomials; selecting algebraic numbers of small degree; sampling extensions via adjoining roots; exploring valued fields at different primes; sampling embeddings; testing random automorphisms; sampling norms/traces of random elements. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Polynomial coefficient lists; factorization outputs; matrices for trace/norm computation; automorphism group elements; valuation tables; discriminant values; extension tower data; embedding maps; root approximations. |
| | Resolution | The granularity or precision with which data is captured. | Determined by precision of polynomial coefficients, fineness of root approximations, depth of extension towers, granularity of valuation scales, and completeness of automorphism enumeration. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Cross-checking polynomial factorization using different algorithms; verifying minimal polynomials; validating Galois group computations; comparing valuations across multiple completions; confirming norm/trace results under different bases; ensuring discriminant consistency. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Incorrect factorization; miscomputed minimal polynomials; wrong extension degrees; mistaken automorphism identifications; numerical errors in root approximations; valuation miscalculations; ramification misclassification; discriminant sign/scale errors. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Polynomial factorization patterns; behavior of roots under field extensions; splitting-field formation; separability/inseparability behavior; degree of extensions; automorphism structure; ramification behavior in valued fields; norm/trace computations; residue-field interactions. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Limited by inability to factor arbitrary polynomials efficiently; difficulty detecting inseparability in large characteristic; computational hardness of Galois group determination; limits in numerically approximating roots; difficulty observing infinite extensions directly. |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Extension degree; characteristic; valuation magnitude; ramification index; discriminant size; residue degree; order of automorphism groups; polynomial degree; norm and trace values; embedding count. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Polynomial factorization algorithms; field arithmetic engines; Galois group solvers; valuation and completion tools; number-field computation systems (PARI/GP, Magma, Sage); discriminant calculators; norm/trace computation tools; root-approximation solvers. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Algebraic element defined as root of a polynomial; separability defined via derivative nonvanishing; extension degree defined by basis size; Galois extension defined as normal + separable; valuation defined as ordered-group mapping; discriminant defined via determinant of trace-forms. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Factoring polynomials; computing minimal polynomials; building extension towers; computing splitting fields; calculating automorphism groups; evaluating norms and traces; computing valuations; performing completions; determining ramification. |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Standardized polynomial factorization workflows; canonical extension-building procedures; structured extraction of Galois groups; fixed protocols for computing discriminants; controlled valuation sampling; consistent norm/trace calculations; uniform tower construction. |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Sampling random polynomials; selecting algebraic numbers of small degree; sampling extensions via adjoining roots; exploring valued fields at different primes; sampling embeddings; testing random automorphisms; sampling norms/traces of random elements. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Polynomial coefficient lists; factorization outputs; matrices for trace/norm computation; automorphism group elements; valuation tables; discriminant values; extension tower data; embedding maps; root approximations. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Determined by precision of polynomial coefficients, fineness of root approximations, depth of extension towers, granularity of valuation scales, and completeness of automorphism enumeration. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Cross-checking polynomial factorization using different algorithms; verifying minimal polynomials; validating Galois group computations; comparing valuations across multiple completions; confirming norm/trace results under different bases; ensuring discriminant consistency. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Incorrect factorization; miscomputed minimal polynomials; wrong extension degrees; mistaken automorphism identifications; numerical errors in root approximations; valuation miscalculations; ramification misclassification; discriminant sign/scale errors. |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Manipulating polynomial inputs to generate different field extensions; adjoining elements to test algebraicity/transcendence; altering valuation parameters; modifying embeddings; constructing different tower configurations; varying coefficients to observe changes in splitting behavior. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observing natural factorization patterns; tracking behavior of automorphisms; monitoring extension degree as roots are adjoined; observing ramification/inertia behavior under prime-specific valuations; watching norm/trace distribution across sampled elements. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Testing whether an element is algebraic by checking polynomial annihilation; testing separability by derivative/nonvanishing criteria; verifying normality via closure under embeddings; testing Galois correspondence predictions; checking whether valuations extend uniquely; verifying ramification indices and residue degrees. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Repeating polynomial factorizations with alternative algorithms; recalculating minimal polynomials; recomputing automorphism groups; repeating discriminant and norm/trace computations across bases; replicating completion and valuation results under multiple software systems. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Analyzing distribution of splitting types for random polynomials; assessing typical extension degrees; evaluating frequency of separable vs. inseparable behavior; comparing norms/traces across random samples; examining distribution of Galois group sizes for classes of polynomials. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Comparing different field models (finite fields vs. number fields vs. function fields); comparing separable vs. inseparable theories; comparing behavior under completions vs. global structures; contrasting tower constructions; evaluating numerical vs. symbolic factorization models. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Detecting incorrect factorizations; miscomputed minimal polynomials; errors in Galois-group algorithms; incorrect valuation assignments; ramification misclassification; numerical instability in root approximations; mistaken norm/trace outputs due to basis errors. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Avoiding bias toward polynomials with “nice” splitting behavior; ensuring randomization in polynomial sampling; controlling for basis choice in norm/trace computations; avoiding representational bias in extension diagrams; controlling computational heuristics. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Verifying correctness of factorization, minimal polynomial, and Galois-group claims; reviewing valuation and ramification calculations; auditing embedding and automorphism results; cross-checking tower and discriminant computations; challenging structural claims about extensions. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Updating extension classifications; refining separability or normality assumptions; modifying valuation frameworks; correcting Galois correspondences; incorporating new results from algebraic number theory or algebraic geometry; adjusting methods for infinite extensions. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Full disclosure of algorithms, monomial orders, factorization methods, valuation rules, basis selections, and assumptions used in extension or Galois computations; explicit reporting of computational limits. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Honest reporting of computational and theoretical results; acknowledging inseparable or pathological counterexamples; ensuring reproducibility of extension and valuation computations; stating undecidability or intractability constraints clearly; maintaining rigor in algebraic claims. |