# Kummer’s lemma

The following result is a key ingredient in the proof of Fermat’s last theorem for regular primes. More concretely, the lemma is needed to show the so-called second case of Fermat, i.e. $x^{p}+y^{p}=z^{p}$ does not have any non-trivial solutions in $\mathbb{Z}$ 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 $\zeta_{p}$ be a primitive $p$th root of unity and let $K=\mathbb{Q}(\zeta_{p})$ be the corresponding cyclotomic field. Let $E$ be the group of algebraic units of the ring of integers $\mathcal{O}_{K}$. Suppose that $p$ is a regular prime. If a unit $\epsilon\in E$ is congruent modulo $p$ to a rational integer, then $\epsilon$ is the $p$th power of another unit also $E$.

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

## References

• Ozaki 1997 Ozaki, M., Kummer’s lemma for $\mathbb{Z}_{p}$-extensions 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 Theory 40 (1992), no. 2, 165–173.
Title Kummer’s lemma KummersLemma 2013-03-22 15:55:21 2013-03-22 15:55:21 alozano (2414) alozano (2414) 6 alozano (2414) Theorem msc 11F80 msc 14H52 msc 11D41