Let $V$ be an algebraic variety defined over a field $K$. By $V(K)$ we denote the set of points on $V$ defined over $K$. Let $\bar{K}$ be an algebraic closure of $K$. For a valuation $\nu$ of $K$, we write $K_{{\nu}}$ for the completion of $K$ at $\nu$. In this case, we can also consider $V$ defined over $K_{{\nu}}$ and talk about $V(K_{{\nu}})$.

The Hasse Principle is the idea (or desire) that an everywhere locally soluble variety $V$ must have a rational point, i.e. a point defined over $K$. Unfortunately this is not true, there are examples of varieties that satisfy the Hasse condition but have no rational points.

Example: A quadric (of any dimension) satisfies the Hasse condition. This was proved by Minkowski for quadrics over $\mathbb{Q}$ and by Hasse for quadrics over a number field.