Atlas of Fundamental Concepts

A number field has many places, each producing a local completion. A global rational point necessarily gives points in every completion, but the converse is a substantive theorem when true. Adeles collect all local fields at once. An everywhere locally soluble variety has a point over each completion. Quadratic forms satisfy the Hasse–Minkowski theorem, a clean local–global principle; other varieties can be everywhere locally soluble yet have no global point, showing why local evidence and global existence must be separated.