Hermite–Minkowski theorem
E502195
The Hermite–Minkowski theorem is a fundamental result in algebraic number theory that gives a finiteness bound on the number of number fields of a given degree and discriminant.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hermite–Minkowski theorem canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T5191869 — 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: Hermite–Minkowski theorem Context triple: [Charles Hermite, hasConceptNamedAfter, Hermite–Minkowski theorem]
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A.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
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B.
Hasse norm theorem
The Hasse norm theorem is a fundamental result in algebraic number theory that characterizes when an element of a global field is a norm from a cyclic extension by relating this property to its behavior in all completions of the field.
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C.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
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D.
Chebotarev density theorem
The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
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E.
Die Theorie der algebraischen Zahlkörper
"Die Theorie der algebraischen Zahlkörper" is a foundational mathematical monograph on algebraic number fields, authored by David Hilbert and published as part of his influential Zahlbericht.
- 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: Hermite–Minkowski theorem Target entity description: The Hermite–Minkowski theorem is a fundamental result in algebraic number theory that gives a finiteness bound on the number of number fields of a given degree and discriminant.
-
A.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
-
B.
Hasse norm theorem
The Hasse norm theorem is a fundamental result in algebraic number theory that characterizes when an element of a global field is a norm from a cyclic extension by relating this property to its behavior in all completions of the field.
-
C.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
D.
Chebotarev density theorem
The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
-
E.
Die Theorie der algebraischen Zahlkörper
"Die Theorie der algebraischen Zahlkörper" is a foundational mathematical monograph on algebraic number fields, authored by David Hilbert and published as part of his influential Zahlbericht.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf | mathematical theorem ⓘ |
| appliesTo |
finite extensions of the rational numbers
ⓘ
number fields of fixed degree over ℚ ⓘ |
| assumes |
bound on the absolute value of the discriminant
ⓘ
fixed degree of the number field ⓘ |
| concerns |
degree of number fields
ⓘ
discriminants of number fields ⓘ finiteness of number fields ⓘ number fields ⓘ |
| context |
arithmetic of algebraic number fields
ⓘ
discriminant bounds for extensions of ℚ ⓘ |
| establishes |
a lower bound on the absolute value of discriminants of nontrivial number fields
ⓘ
that only finitely many number fields have discriminant bounded by a fixed constant ⓘ |
| field | algebraic number theory ⓘ |
| formalizes | finiteness of number fields with bounded invariants ⓘ |
| gives | explicit upper bounds on discriminants of number fields ⓘ |
| hasConsequence |
finiteness of algebraic integers of bounded degree and bounded discriminant
ⓘ
finiteness of class of orders in number fields with bounded discriminant ⓘ only finitely many isomorphism classes of number fields of given degree and bounded discriminant ⓘ |
| hasProofMethod |
geometric arguments in Euclidean space
ⓘ
volume estimates of convex symmetric bodies ⓘ |
| holdsOver | the rational number field ℚ ⓘ |
| implies |
there are only finitely many number fields of given degree and bounded discriminant
ⓘ
there are only finitely many number fields of given degree and given discriminant bound ⓘ |
| involves |
discriminant as covolume of a lattice
ⓘ
embedding of number fields into ℝⁿ ⓘ ring of integers of a number field ⓘ |
| isClassicalResult | 19th-century number theory ⓘ |
| isPartOf | classical results in algebraic number theory ⓘ |
| namedAfter |
Charles Hermite
ⓘ
Hermann Minkowski ⓘ |
| relatedTo |
Hermite’s constant
ⓘ
linked to:
Hermite constant
Minkowski’s theorem ⓘ
linked to:
Minkowski’s theorem on convex sets
geometry of numbers ⓘ |
| relates | degree of a number field to its discriminant ⓘ |
| typeOfResult |
discriminant bound
ⓘ
finiteness theorem ⓘ |
| usedFor |
bounding the number of extensions of ℚ with given properties
ⓘ
effective enumeration of number fields of small discriminant ⓘ |
| usedIn |
classification of number fields
ⓘ
effective results in arithmetic statistics ⓘ proofs of finiteness of class numbers in certain settings ⓘ |
| uses |
Minkowski’s convex body theorem
ⓘ
linked to:
Minkowski’s theorem on convex sets
geometry of numbers ⓘ lattice point counting ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Hermite–Minkowski theorem Description of subject: The Hermite–Minkowski theorem is a fundamental result in algebraic number theory that gives a finiteness bound on the number of number fields of a given degree and discriminant.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.