Hasse norm theorem

E213039

result in algebraic number theory 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.

Try in SPARQL Jump to: Surface forms Statements Referenced by

All labels observed (2)

Label	Occurrences
Hasse norm theorem canonical	1
for a cyclic extension L/K of global fields, an element of K is a global norm from L if and only if it is a local norm at every place of K	1

Statements (43)

Predicate	Object
instanceOf	result in algebraic number theory ⓘ theorem ⓘ
appliesTo	cyclic extensions of global fields ⓘ function fields of one variable over finite fields ⓘ global fields ⓘ number fields ⓘ
assumes	K is a global field ⓘ L over K is a finite cyclic extension ⓘ
characterizes	norms from cyclic extensions ⓘ
concerns	elements that are everywhere local norms ⓘ
conclusion	local norm conditions are sufficient for global norm representation in cyclic extensions ⓘ
failsInGeneralFor	non-cyclic extensions ⓘ
field	algebraic number theory ⓘ
generalizes	Hasse principle for norms in cyclic extensions ⓘ
hasFormulationIn	Galois cohomology ⓘ cohomological terms ⓘ idele-theoretic language ⓘ
historicalPeriod	20th century mathematics ⓘ
holdsFor	cyclic Galois extensions ⓘ
involves	global class field theory ⓘ idele class groups ⓘ idele groups ⓘ local class field theory ⓘ local completions of global fields ⓘ norm map from an extension field to its base field ⓘ
namedAfter	Helmut Hasse ⓘ
provenBy	Helmut Hasse ⓘ
relatedTo	Artin reciprocity law ⓘ Hasse principle ⓘ Herbrand quotient ⓘ Hilbert symbol ⓘ surface form: Hilbert reciprocity law Shafarevich group of a torus ⓘ Tate cohomology ⓘ global reciprocity map ⓘ
relates	global norm conditions to local norm conditions ⓘ
states	Hasse norm theorem self-linksurface differs ⓘ surface form: for a cyclic extension L/K of global fields, an element of K is a global norm from L if and only if it is a local norm at every place of K
usedIn	analysis of weak approximation on norm varieties ⓘ arithmetic of algebraic tori ⓘ class field theory ⓘ computations of relative Brauer groups ⓘ description of norm groups in cyclic extensions ⓘ proofs of local-global principles ⓘ study of norm one tori ⓘ

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: Hasse norm theorem
Description of subject: 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.

Referenced by (2)

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

Helmut Hasse → notableWork → Hasse norm theorem ⓘ

Hasse norm theorem → states → Hasse norm theorem self-linksurface differs ⓘ

this entity surface form: for a cyclic extension L/K of global fields, an element of K is a global norm from L if and only if it is a local norm at every place of K