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's theorem (number theory) | Statement: [Kummer, hasEponym, Kummer's theorem (number theory)]

Target entity: Kummer's theorem (number theory)
Context triple: [Kummer, hasEponym, Kummer's theorem (number theory)]

• A. Kummer congruences
  Kummer congruences are number-theoretic relations describing how special values of Bernoulli numbers and related arithmetic functions behave modulo powers of primes, foundational in the study of p-adic L-functions and cyclotomic fields.
• B. 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.
• C. Gauss’s lemma in number theory
  Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
• D. Ribet's theorem
  Ribet's theorem is a result in number theory that linked certain modular forms to Galois representations and played a crucial role in the proof of Fermat's Last Theorem.
• E. Fermat's theorem on sums of two squares
  Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Kummer's theorem (number theory)
Target entity description: Kummer's theorem (number theory) is a result that characterizes the highest power of a prime dividing a binomial coefficient by counting the carries that occur when adding the binomial parameters in base that prime.

• A. Kummer congruences
  Kummer congruences are number-theoretic relations describing how special values of Bernoulli numbers and related arithmetic functions behave modulo powers of primes, foundational in the study of p-adic L-functions and cyclotomic fields.
• B. 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.
• C. Gauss’s lemma in number theory
  Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
• D. Ribet's theorem
  Ribet's theorem is a result in number theory that linked certain modular forms to Galois representations and played a crucial role in the proof of Fermat's Last Theorem.
• E. Fermat's theorem on sums of two squares
  Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
• F. None of above. chosen