Triple

T10063322

Position	Surface form	Disambiguated ID	Type / Status
Subject	Hasse norm theorem	`E213039`	entity
Predicate	states	`P34`	FINISHED
Object	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	`E213039`	NE FINISHED

Named-entity recognition

Before disambiguation, gpt-5-mini classified whether the object phrase is a named entity — the step behind the object's NE type shown above.

Instruction

Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).

Input

Phrase: 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 | Statement: [Hasse norm theorem, states, 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]

Disambiguation candidates (1 decision)

The exact options the model was shown at each disambiguation step, with the option it chose highlighted — the evidence behind this triple's disambiguated ids.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: 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
Context triple: [Hasse norm theorem, states, 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]

A. Hasse norm theorem chosen
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.
B. Kronecker–Weber theorem
The Kronecker–Weber theorem is a fundamental result in algebraic number theory stating that every finite abelian extension of the rational numbers is contained in a cyclotomic field generated by roots of unity.
C. Artin reciprocity law
The Artin reciprocity law is a fundamental theorem in class field theory that generalizes quadratic reciprocity by describing abelian extensions of number fields in terms of characters of their idele class groups.
D. Frobenius element
The Frobenius element is a distinguished element in a Galois group associated to an unramified prime, encoding how that prime splits in a field extension and playing a central role in algebraic number theory and arithmetic geometry.
E. Furtwängler’s theorem in class field theory
Furtwängler’s theorem in class field theory is a fundamental result in algebraic number theory that refines the principal ideal theorem by describing how ideal classes capitulate (become principal) in certain abelian extensions of number fields.
F. None of above.
G. Unsure - the case is ambiguous/there is not enough information to decide.

Provenance (3 batches)

Stage	Batch ID	Job type	Status
creating	`batch_69ca83977128819084084eb7d1d8c52a`	elicitation	completed
NER	`batch_69cdcfd4e4ac8190a37061b4082caa48`	ner	completed
NED1	`batch_69d29a7bd56c8190a6c43df26db880f4`	ned_source_triple	completed

Created at: March 30, 2026, 8:58 p.m.