Hasse Principle

A collection of equations satisfies the Hasse principle if, whenever one of the equations has solutions in $\Bbb{R}$ and all the $\Bbb{Q}_p$, then the equations have solutions in the Rationals $\Bbb{Q}$. Examples include the set of equations

$$ax^2 + bxy + cy^2 = 0$$

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.

See also Local Field