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 CyclotomicFields, 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.

单词	KummersLemma
释义	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 CyclotomicFields, 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.
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