Soit une extension finie de d’indice de ramification , et soit une -extension abélienne finie de groupe de Galois et d’indice de ramification . Nous donnons un critère en termes des nombres de ramification permettant de décider lorsqu’un idéal fractionnaire de l’anneau de valuation de peut être libre sur son ordre associé . En particulier, si , la codifférente ne peut être libre sur son ordre associé que si (mod ) pour tout . Nous déduisons de cela trois conséquences. Premièrement, si est un ordre de Hopf et si est une -extension galoisienne, où est l’anneau de valuation de , alors (mod ) pour tout . Deuxièmement, si et sont des corps de points de division d’un groupe de Lubin-Tate, avec et , alors n’est pas libre sur . Troisièmement, ces extensions possèdent deux structures galoisiennes de Hopf différentes, mettant en évidence des comportements différents au niveau des entiers.
Let be a finite extension of with ramification index , and let be a finite abelian -extension with Galois group and ramification index . We give a criterion in terms of the ramification numbers for a fractional ideal of the valuation ring of not to be free over its associated order . In particular, if then the inverse different can be free over its associated order only when (mod ) for all . We give three consequences of this. Firstly, if is a Hopf order and is -Galois then (mod ) for all . Secondly, if are Lubin-Tate division fields, with and , then is not free over (. Thirdly, these extensions admit two Hopf Galois structures exhibiting different behaviour at integral level.
Classification : 11S23, 11R33, 11S31, 16W30
Mots clés : Galois module structure, associated order, Hopf order, Lubin-Tate formal group
@article{JTNB_1997__9_1_201_0, author = {Byott, Nigel P.}, title = {Galois structure of ideals in wildly ramified abelian $p$-extensions of a $p$-adic field, and some applications}, journal = {Journal de Th\'eorie des Nombres de Bordeaux}, pages = {201--219}, publisher = {Universit\'e Bordeaux I}, volume = {9}, number = {1}, year = {1997}, doi = {10.5802/jtnb.196}, zbl = {0889.11040}, mrnumber = {1469668}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.196/} }
Nigel P. Byott. Galois structure of ideals in wildly ramified abelian $p$-extensions of a $p$-adic field, and some applications. Journal de Théorie des Nombres de Bordeaux, Tome 9 (1997) no. 1, pp. 201-219. doi : 10.5802/jtnb.196. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.196/
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