Kummer's lemma

Target entity: Kummer's lemma
Context triple: [Kummer, hasEponym, Kummer's lemma]

• A. Kummer's theorem (number theory)
  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.
• B. 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.
• C. Kronecker’s lemma
  Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
• D. 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.
• E. Hensel’s lemma
  Hensel’s lemma is a fundamental result in number theory and p-adic analysis that allows one to lift solutions of polynomial congruences modulo a prime power to higher powers, analogous to Newton’s method in the p-adic setting.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Kummer's lemma
Target entity description: 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.

• A. Kummer's theorem (number theory)
  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.
• B. 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.
• C. Kronecker’s lemma
  Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
• D. 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.
• E. Hensel’s lemma
  Hensel’s lemma is a fundamental result in number theory and p-adic analysis that allows one to lift solutions of polynomial congruences modulo a prime power to higher powers, analogous to Newton’s method in the p-adic setting.
• F. None of above. chosen