Hermite–Minkowski theorem

E502195

mathematical theorem

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.

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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 NERFINISHED ⓘ Hermann Minkowski NERFINISHED ⓘ
relatedTo	Hermite’s constant NERFINISHED ⓘ Minkowski’s theorem NERFINISHED ⓘ geometry of numbers NERFINISHED ⓘ
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 NERFINISHED ⓘ geometry of numbers ⓘ lattice point counting ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Charles Hermite → hasConceptNamedAfter → Hermite–Minkowski theorem ⓘ