local-global principle

A collection of equations satisfies the Hasse principle if, whenever one of the equations has solutions in R and all the Q_p, then the equations have solutions in the rationals Q. Examples include the set of equations a x^2 + b x y + c y^2 = 0 with a, b, and c integers, and the set of equations x^2 + y^2 = a for a rational. The trivial solution x = y = 0 is usually not taken into account when deciding if a collection of homogeneous equations satisfies the Hasse principle. The Hasse principle is sometimes called the local-global principle.

global field | local field