View source for Place (Mathematics)

In mathematics, a '''place''' of a field is an equivalence class of absolute values, or equivalently, a completion of the field at an absolute value. For an [[Algebraic Number Field|algebraic number field]], the places correspond to embeddings into '''R''' or '''C''' (the infinite places) and to nonzero prime ideals of the ring of integers (the finite places). The finite places yield the ''p''-adic fields, and together the places encode the full arithmetic-geometric structure of the field. The collection of all places is the foundation of the '''[[Adele Ring|adele ring]]''', which unifies the local completions into a single global object.\n\n''A place is not a point in any naive sense. It is a lens — a way of looking at a field that reveals one aspect of its structure while hiding others. The miracle of modern number theory is that the field can be reconstructed from its lenses, through the geometry of the '''[[Adele Ring|adele ring]]''' and the theorems of class field theory.''\n\n[[Category:Mathematics]]