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Kummer's lemma
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(Theorem)
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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 $\Ints$ with $p>2$ a regular prime and $p|xyz$ . It is due to Ernst Kummer, thus the name.
For a proof, see [Washington], Theorem 5.36. The reader may also be interested in generalizations due to [Washington 1992] and [Ozaki 1997].
- Ozaki 1997
- Ozaki, M., Kummer's lemma for $\Ints_p$ -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.
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"Kummer's lemma" is owned by alozano.
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Cross-references: theorem, rational integer, congruent, unit, ring of integers, algebraic units, group, cyclotomic field, root of unity, primitive, prime, solutions, regular primes, Fermat's last theorem, proof, key
This is version 3 of Kummer's lemma, born on 2006-05-26, modified 2006-09-26.
Object id is 7929, canonical name is KummersLemma.
Accessed 1464 times total.
Classification:
| AMS MSC: | 11D41 (Number theory :: Diophantine equations :: Higher degree equations; Fermat's equation) | | | 14H52 (Algebraic geometry :: Curves :: Elliptic curves) | | | 11F80 (Number theory :: Discontinuous groups and automorphic forms :: Galois representations) |
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Pending Errata and Addenda
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