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.
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}, mrnumber = {1469668}, zbl = {0889.11040}, language = {en}, url = {http://archive.numdam.org/item/JTNB_1997__9_1_201_0/} }
TY - JOUR AU - Byott, Nigel P. TI - Galois structure of ideals in wildly ramified abelian $p$-extensions of a $p$-adic field, and some applications JO - Journal de théorie des nombres de Bordeaux PY - 1997 SP - 201 EP - 219 VL - 9 IS - 1 PB - Université Bordeaux I UR - http://archive.numdam.org/item/JTNB_1997__9_1_201_0/ LA - en ID - JTNB_1997__9_1_201_0 ER -
%0 Journal Article %A Byott, Nigel P. %T Galois structure of ideals in wildly ramified abelian $p$-extensions of a $p$-adic field, and some applications %J Journal de théorie des nombres de Bordeaux %D 1997 %P 201-219 %V 9 %N 1 %I Université Bordeaux I %U http://archive.numdam.org/item/JTNB_1997__9_1_201_0/ %G en %F JTNB_1997__9_1_201_0
Byott, Nigel P. 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. http://archive.numdam.org/item/JTNB_1997__9_1_201_0/
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