SWhat changed
Steinitz organizes abstract commutative fields using prime fields, algebraic closure and transcendence bases, and classifies algebraically closed fields by characteristic and transcendence degree.
Scientific scope
Steinitz organizes abstract commutative fields using prime fields, algebraic closure and transcendence bases, and classifies algebraically closed fields by characteristic and transcendence degree. Scope is the original 1910 commutative-field structure framework described on printed pages 167–186 and 279–309. Every field has an algebraically closed algebraic extension, unique up to base-fixing isomorphism, not a unique chosen map. Transcendence bases and their cardinal invariant use choice/well-ordering and earlier within-paper lemmas; the intervening lemmas on printed pages 187–278 have not been independently checked for this record. Characteristic and transcendence degree classify algebraically closed fields up to abstract isomorphism; they do not classify all fields. General fields occupy intermediate extensions between a rational-function field and an algebraic closure. No uniform effective algorithm is supplied. The original says the higher-transcendence-degree analogue of Lüroth remains unresolved; the one-variable result is not extrapolated. Weber's field definition and preceding root-adjunction methods are inherited. No theorem equating closure existence with choice in ZF is claimed. The complete proof chain has not been independently reconstructed. Further perfect/imperfect-field details are outside the assessed contribution scope.
Evidence
Source-linked, scoped account of the original work. The published scope states the scientific assumptions and remaining verification limits. WRS has not independently certified the full proof, experimental result or system benchmark.
Contributors
- Ernst Steinitz · SCIENCE: Original source-listed scientific author of this scoped work; byline does not determine effort fractions.
- Ernst Steinitz, Algebraische Theorie der Körper (1910), original GDZ article facsimile
- Mathdoc original-article bibliography: Steinitz, Journal für die reine und angewandte Mathematik 137 (1910), 167–309
Provisional WRS assessment, edition 2026-10-16.1. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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