What is recorded here?
Steinitz organizes abstract commutative fields using prime fields, algebraic closure and transcendence bases, and classifies algebraically closed fields by characteristic and transcendence degree. A field may look like rational numbers, finite arithmetic or rational functions, yet the same abstract operations can govern it. Steinitz separates two ways to enlarge it: add algebraically independent quantities, then add roots of polynomial equations. This exposes reusable structure across different examples. His framework guarantees algebraic closures and gives an invariant size to a transcendence basis. The particularly clean classification by characteristic and transcendence degree applies to algebraically closed fields; ordinary fields with those same invariants can still differ. The existence proofs use well-ordering rather than providing a universal computer procedure.
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Scientific scope
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.
Read the original work
- 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
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People and systems
Ernst Steinitz
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English catalogue narrative
Steinitz: organizing abstract fields through algebraic closure and transcendence
Steinitz organizes abstract commutative fields using prime fields, algebraic closure and transcendence bases, and classifies algebraically closed fields by characteristic and transcendence degree.
A field may look like rational numbers, finite arithmetic or rational functions, yet the same abstract operations can govern it. Steinitz separates two ways to enlarge it: add algebraically independent quantities, then add roots of polynomial equations. This exposes reusable structure across different examples. His framework guarantees algebraic closures and gives an invariant size to a transcendence basis. The particularly clean classification by characteristic and transcendence degree applies to algebraically closed fields; ordinary fields with those same invariants can still differ. The existence proofs use well-ordering rather than providing a universal computer procedure.
Scientific scope
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.
People and systems
Ernst Steinitz