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: Noether's problem over the rational numbers | Statement: [Noether's problem, hasSpecialCase, Noether's problem over the rational numbers]

Target entity: Noether's problem over the rational numbers
Context triple: [Noether's problem, hasSpecialCase, Noether's problem over the rational numbers]

• A. Noether's problem chosen
  Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
• B. Hilbert’s irreducibility theorem
  Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
• C. Noether's isomorphism theorems
  Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
• D. Hilbert’s Nullstellensatz
  Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
• E. Swan constructed counterexamples over the rational numbers
  Swan constructed counterexamples over the rational numbers refers to Richard G. Swan’s landmark result showing that certain invariant fields under finite group actions over the rational numbers are not rational, thereby disproving a general affirmative answer to Noether’s problem in this setting.
• F. None of above.
• G. Unsure - the case is ambiguous/there is not enough information to decide.