Definitions
The profinite completion of the integers, Ẑ, is the inverse limit of the rings Z/nZ:
By the Chinese remainder theorem it is isomorphic to the product of all the rings of p-adic integers:
The ring of integral adeles AZ is the product
The ring of (rational) adeles AQ is the tensor product
(topologized so that AZ is an open subring).
More generally the ring of adeles AF of any algebraic number field F is the tensor product
(topologized as the product of copies of AQ).
The ring of (rational) adeles can also be defined as the restricted product
of all the p-adic completions Qp and the real numbers (or in other words as the restricted product of all completions of the rationals). In this case the restricted product means that for an adele (a∞, a2, a3, a5, …) all but a finite number of the ap are p-adic integers.
The adeles of a function field over a finite field can be defined in a similar way, as the restricted product of all completions.
Read more about this topic: Adele Ring
Famous quotes containing the word definitions:
“The loosening, for some people, of rigid role definitions for men and women has shown that dads can be great at calming babies—if they take the time and make the effort to learn how. It’s that time and effort that not only teaches the dad how to calm the babies, but also turns him into a parent, just as the time and effort the mother puts into the babies turns her into a parent.”
—Pamela Patrick Novotny (20th century)
“Lord Byron is an exceedingly interesting person, and as such is it not to be regretted that he is a slave to the vilest and most vulgar prejudices, and as mad as the winds?
There have been many definitions of beauty in art. What is it? Beauty is what the untrained eyes consider abominable.”
—Edmond De Goncourt (1822–1896)
“What I do not like about our definitions of genius is that there is in them nothing of the day of judgment, nothing of resounding through eternity and nothing of the footsteps of the Almighty.”
—G.C. (Georg Christoph)