Let be a number field. For each finite prime of , let be the valuation ring of the completion of at , and let be the group of units in . Then each group is a compact open subgroup of the group of units of . The idèle group of is defined to be the restricted direct product of the multiplicative groups with respect to the compact open subgroups , taken over all finite primes and infinite primes of .
The units in embed into via the diagonal embedding
where is the image of under the embedding of into its completion . As in the case of adèles, the group is a discrete subgroup of the group of idèles , but unlike the case of adèles, the quotient group is not a compact group. It is, however, possible to define a certain subgroup of the idèles (the subgroup of norm 1 elements) which does have compact quotient under .
|Date of creation||2013-03-22 12:39:28|
|Last modified on||2013-03-22 12:39:28|
|Last modified by||djao (24)|
|Defines||group of idèles|