A simple proof of Noether's theorem
Glasgow mathematical journal, Tome 38 (1996) no. 1, pp. 49-51
Voir la notice de l'article provenant de la source Cambridge University Press
We present an elementary proof of the theorem, usually attributed to Noether, that if L/K is a tame finite Galois extension of local fields, then is a free -module where Γ=Gal(L/K. The attribution to Noether is slightly misleading as she only states and proves the result in the case where the residual characteristic of K does not divide the order of Γ [4]. In this case is a maximal order in KΓ which is not true for general groups Γ. There is an elegant proof in the standard reference [2], but this relies on a difficult result in representation theory due to Swan. Our proof depends on a close examination of the structure of tame local extensions, and uses only elementary facts about local fields. It also gives an explicit construction of a generator element, and the same proof works both for localizations of number fields and of global function fields.
Chapman, Robin J. A simple proof of Noether's theorem. Glasgow mathematical journal, Tome 38 (1996) no. 1, pp. 49-51. doi: 10.1017/S0017089500031244
@article{10_1017_S0017089500031244,
author = {Chapman, Robin J.},
title = {A simple proof of {Noether's} theorem},
journal = {Glasgow mathematical journal},
pages = {49--51},
year = {1996},
volume = {38},
number = {1},
doi = {10.1017/S0017089500031244},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500031244/}
}
[1] 1.Cohn, P. M., Algebra, vol. 3 (2nd ed.) (Wiley, 1991). Google Scholar
[2] 2.Fröhlich, A., Galois module structure of algebraic integers (Springer, 1983). Google Scholar | DOI
[3] 3.Matsumura, H., Commutative ring theory (Cambridge University Press, 1986). Google Scholar
[4] 4.Noether, E., Normalbasis bei Körpern ohne höhere Verzweigung, J. Reine Angew. Math. 167 (1932), 147–152. Google Scholar
[5] 5.Serre, J.-P., Local fields (Springer, 1979). Google Scholar
Cité par Sources :