Levi-Civita field
E627727
The Levi-Civita field is a non-Archimedean ordered field of formal power series with real coefficients and well-ordered rational exponents, used to rigorously model infinitesimals and infinite quantities in analysis.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Levi-Civita field canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6910737 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Levi-Civita field Context triple: [Hahn series, relatedConcept, Levi-Civita field]
-
A.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
-
B.
Levi-Civita symbol
The Levi-Civita symbol is an antisymmetric tensor used in mathematics and physics to represent orientations, cross products, and determinants in multiple dimensions.
-
C.
Infeld–van der Waerden formalism
The Infeld–van der Waerden formalism is a mathematical framework in general relativity that reformulates the theory using spinor calculus to describe gravitational and electromagnetic fields.
-
D.
Noether field
A Noether field is a type of field extension studied in invariant theory and Galois theory, arising as the fixed field of a group action on a rational function field and central to questions about rationality in Noether’s problem.
-
E.
Ricci-Curbastro
Ricci-Curbastro is the surname of the Italian mathematician Gregorio Ricci-Curbastro, a pioneer of tensor calculus and differential geometry.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Levi-Civita field Target entity description: The Levi-Civita field is a non-Archimedean ordered field of formal power series with real coefficients and well-ordered rational exponents, used to rigorously model infinitesimals and infinite quantities in analysis.
-
A.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
-
B.
Levi-Civita symbol
The Levi-Civita symbol is an antisymmetric tensor used in mathematics and physics to represent orientations, cross products, and determinants in multiple dimensions.
-
C.
Infeld–van der Waerden formalism
The Infeld–van der Waerden formalism is a mathematical framework in general relativity that reformulates the theory using spinor calculus to describe gravitational and electromagnetic fields.
-
D.
Noether field
A Noether field is a type of field extension studied in invariant theory and Galois theory, arising as the fixed field of a group action on a rational function field and central to questions about rationality in Noether’s problem.
-
E.
Ricci-Curbastro
Ricci-Curbastro is the surname of the Italian mathematician Gregorio Ricci-Curbastro, a pioneer of tensor calculus and differential geometry.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
Hahn field
ⓘ
non-Archimedean field ⓘ ordered field ⓘ real-closed field ⓘ valued field ⓘ |
| contains | infinitesimal neighborhoods of real numbers ⓘ |
| extends | field of real numbers ⓘ |
| hasAnalyticUse | definition of derivatives via infinitesimal quotients ⓘ |
| hasAnalyticUse | definition of integrals via infinitesimal partitions ⓘ |
| hasArchimedeanProperty | fails the Archimedean property ⓘ |
| hasCharacteristic | 0 ⓘ |
| hasCoefficientField | real numbers ⓘ |
| hasComparison |
is different from Robinson’s hyperreal field
ⓘ
provides an alternative to nonstandard analysis ⓘ |
| hasConstruction | formal power series with real coefficients and well-ordered rational exponents ⓘ |
| hasElementForm | sum over q in Q of a_q t^q with well-ordered support ⓘ |
| hasElements | formal power series ⓘ |
| hasExponentCondition | supports are well-ordered subsets of the rationals ⓘ |
| hasExponentGroup | rational numbers ⓘ |
| hasInfiniteElementProperty | every infinitely large element is larger than any real number ⓘ |
| hasInfinitesimal | t with 0 < t < 1/n for all positive integers n ⓘ |
| hasInfinitesimalProperty | every positive infinitesimal is smaller than any positive real number ⓘ |
| hasOrdering | lexicographic order on exponents and coefficients ⓘ |
| hasProperty |
Cauchy complete with respect to its natural valuation
ⓘ
contains infinitely large elements ⓘ contains infinitesimal elements ⓘ isomorphic copies of the real numbers embed as constant series ⓘ non-Archimedean ⓘ real-closed ⓘ spherically complete ⓘ strictly extends the real numbers ⓘ totally ordered ⓘ |
| hasSeriesSupportProperty | each nonzero element has a least exponent in its support ⓘ |
| hasSubstructure | copy of the rational numbers as constant series with rational coefficients ⓘ |
| hasTopology | valuation topology induced by its natural valuation ⓘ |
| hasUse |
asymptotic analysis
ⓘ
generalized differential calculus ⓘ non-Archimedean functional analysis ⓘ nonstandard-style analysis without ultrafilters ⓘ rigorous modeling of infinite quantities ⓘ rigorous modeling of infinitesimals ⓘ |
| hasValuation | map sending nonzero series to least exponent with nonzero coefficient ⓘ |
| isSubfieldOf | field of Hahn series with real coefficients and rational exponents ⓘ |
| namedAfter | Tullio Levi-Civita ⓘ |
| supports |
Cauchy product of series for multiplication
ⓘ
termwise addition of series ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Levi-Civita field Description of subject: The Levi-Civita field is a non-Archimedean ordered field of formal power series with real coefficients and well-ordered rational exponents, used to rigorously model infinitesimals and infinite quantities in analysis.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.