Hasse principle

Summary

The Hasse principle, or local-global principle, concerns polynomial equations with rational coefficients. It says that such an equation should have a rational solution when it has a real solution and a p-adic solution for each prime $p$. Local solvability is easy to test, so when the principle holds it turns a hard question into finitely many checks. The Hasse–Minkowski theorem shows that it holds for quadratic forms. It fails for some cubic equations, and the failures are measured by objects such as the Tate–Shafarevich group.