Kummer’s lemma

The following result is a key ingredient in the proof of Fermat’s last theoremMathworldPlanetmath for regular primesMathworldPlanetmath. More concretely, the lemma is needed to show the so-called second case of Fermat, i.e. xp+yp=zp does not have any non-trivial solutions in with p>2 a regular prime and p|xyz. It is due to Ernst Kummer, thus the name.

Theorem (Kummer’s Lemma).

Let p>2 be a prime, let ζp be a primitive pth root of unityMathworldPlanetmath and let K=Q(ζp) be the corresponding cyclotomic fieldMathworldPlanetmath. Let E be the group of algebraic units of the ring of integers OK. Suppose that p is a regular prime. If a unit ϵE is congruent modulo p to a rational integer, then ϵ is the pth power of another unit also E.

For a proof, see [Washington], Theorem 5.36. The reader may also be interested in generalizationsPlanetmathPlanetmath due to [Washington 1992] and [Ozaki 1997].


  • Ozaki 1997 Ozaki, M., Kummer’s lemma for Zp-extensionsPlanetmathPlanetmath over totally real number fields, Acta Arith. 81 (1997), no. 1, 37–44.
  • Washington Washington L. C., Introduction to Cyclotomic Fields, Second Edition, Springer-Verlag, New York.
  • Washington 1992 Washington, L. C., Kummer’s lemma for prime power cyclotomic fields, J. Number TheoryMathworldPlanetmathPlanetmath 40 (1992), no. 2, 165–173.
Title Kummer’s lemma
Canonical name KummersLemma
Date of creation 2013-03-22 15:55:21
Last modified on 2013-03-22 15:55:21
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 6
Author alozano (2414)
Entry type Theorem
Classification msc 11F80
Classification msc 14H52
Classification msc 11D41