Kummer theory

E463784

branch of algebraic number theory theory in field theory

Kummer theory is a branch of algebraic number theory that studies abelian extensions of fields, especially cyclotomic and radical extensions, using properties of roots of unity and ideal class groups.

Try in SPARQL Jump to: Surface forms Statements Referenced by

Observed surface forms (1)

Surface form	Occurrences
Kummer extensions	1

Statements (49)

Predicate	Object
instanceOf	branch of algebraic number theory ⓘ theory in field theory ⓘ
appliesTo	global fields ⓘ local fields ⓘ number fields ⓘ
assumes	base field contains enough roots of unity ⓘ
characterizes	Galois group as subgroup of multiplicative group modulo n-th powers ⓘ abelian extensions via n-th power classes ⓘ
describes	structure of abelian extensions of prime power degree ⓘ
extends	Gauss’s theory of quadratic residues NERFINISHED ⓘ
focusesOn	abelian extensions of exponent n ⓘ
formalism	often expressed using Galois cohomology groups H^1 and H^2 ⓘ
generalizes	classical theory of cyclotomic fields ⓘ
hasGeneralization	Artin reciprocity NERFINISHED ⓘ Iwasawa theory NERFINISHED ⓘ Lubin–Tate theory NERFINISHED ⓘ
hasVariant	global Kummer theory ⓘ local Kummer theory ⓘ
historicalContext	developed in the 19th century ⓘ
influenced	development of class field theory ⓘ development of ideal theory ⓘ
isPartOf	abelian extension theory ⓘ algebraic number theory ⓘ
namedAfter	Ernst Kummer NERFINISHED ⓘ
originallyMotivatedBy	study of Fermat's Last Theorem ⓘ study of cyclotomic fields ⓘ
prefigures	global class field theory NERFINISHED ⓘ
relatedConcept	Kummer character ⓘ Kummer extension NERFINISHED ⓘ Kummer radical ⓘ Kummer surface (terminologically related but different area) ⓘ
relatesTo	Galois cohomology NERFINISHED ⓘ Hilbert class field NERFINISHED ⓘ Kummer map NERFINISHED ⓘ Kummer pairing NERFINISHED ⓘ Kummer sequence NERFINISHED ⓘ class field theory NERFINISHED ⓘ ideal class group ⓘ ray class group ⓘ
requires	primitive n-th root of unity in the base field ⓘ
studies	abelian extensions of fields ⓘ cyclotomic extensions ⓘ radical extensions ⓘ
typicalAssumption	characteristic of the field does not divide n ⓘ
uses	ideal class groups ⓘ local–global principles ⓘ multiplicative group of a field ⓘ n-th power residue symbol ⓘ roots of unity ⓘ

Referenced by (5)

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

cyclotomic fields → centralTo → Kummer theory ⓘ

subject surface form: cyclotomic field

Ernst Eduard Kummer → knownFor → Kummer theory ⓘ

this entity surface form: Kummer extensions

Gaussian periods → relatedTo → Kummer theory ⓘ

Cohomologie Galoisienne → topic → Kummer theory ⓘ