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 $p\mathrm{>}\mathrm{2}$ be a prime, let ${\zeta}_{p}$ be a primitive $p$th root of unity^{} and let $K\mathrm{=}\mathrm{Q}\mathit{}\mathrm{(}{\zeta}_{p}\mathrm{)}$ be the corresponding cyclotomic field^{}. Let $E$ be the group of algebraic units of the ring of integers ${\mathrm{O}}_{K}$. Suppose that $p$ is a regular prime. If a unit $\u03f5\mathrm{\in}E$ is congruent modulo $p$ to a rational integer, then $\u03f5$ 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 ${\mathrm{Z}}_{p}$-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 |
---|---|
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 |