Triple

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).

Phrase: Kummer extension | Statement: [Kummer theory, relatedConcept, Kummer extension]

Target entity: Kummer extension
Context triple: [Kummer theory, relatedConcept, Kummer extension]

• A. Kummer 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.
• B. Galois extension
  A Galois extension is a field extension that is both normal and separable, characterized by a well-structured group of automorphisms known as its Galois group.
• C. Kummer's lemma
  Kummer's lemma is a result in algebraic number theory that provides conditions on ideal factorization in cyclotomic fields, particularly relating to divisibility properties of class numbers and units.
• D. Kummer sequence
  The Kummer sequence is a short exact sequence in algebraic number theory and algebraic geometry that relates a ring or scheme to its group of roots of unity and underpins Kummer theory and the study of cyclic extensions.
• E. 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.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Kummer extension
Target entity description: A Kummer extension is a type of field extension obtained by adjoining roots of elements whose powers lie in the base field, playing a central role in the study of abelian extensions in number theory and algebraic geometry.

• A. Kummer theory chosen
  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.
• B. Galois extension
  A Galois extension is a field extension that is both normal and separable, characterized by a well-structured group of automorphisms known as its Galois group.
• C. Kummer's lemma
  Kummer's lemma is a result in algebraic number theory that provides conditions on ideal factorization in cyclotomic fields, particularly relating to divisibility properties of class numbers and units.
• D. Kummer sequence
  The Kummer sequence is a short exact sequence in algebraic number theory and algebraic geometry that relates a ring or scheme to its group of roots of unity and underpins Kummer theory and the study of cyclic extensions.
• E. 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.
• F. None of above.