Hasse principle

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 K¯ be an algebraic closureMathworldPlanetmath of K. For a valuationMathworldPlanetmath ν of K, we write Kν for the completion of K at ν. In this case, we can also consider V defined over Kν and talk about V(Kν).

Definition 1.
  1. 1.

    If V(K) is not empty we say that V is soluble in K.

  2. 2.

    If V(Kν) is not empty then we say that V is locally soluble at ν.

  3. 3.

    If V is locally soluble for all ν then we say that V satisfies the Hasse condition, or we say that V/K is everywhere locally soluble.

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 and by Hasse for quadrics over a number fieldMathworldPlanetmath.


  • 1 Swinnerton-Dyer, Diophantine EquationsMathworldPlanetmath: Progress and Problems, http://swc.math.arizona.edu/notes/files/DLSSw-Dyer1.pdfonline notes.
Title Hasse principle
Canonical name HassePrinciple
Date of creation 2013-03-22 13:50:39
Last modified on 2013-03-22 13:50:39
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 7
Author alozano (2414)
Entry type Definition
Classification msc 14G05
Related topic HasseMinkowskiTheorem
Defines Hasse principle
Defines Hasse condition
Defines locally soluble