Hasse-Minkowski theorem

The Hasse-Minkowski theoremMathworldPlanetmath is a classical example of the Hasse principleMathworldPlanetmath.

Let F be a global fieldMathworldPlanetmath, i.e. a number fieldMathworldPlanetmath or a rational function fieldPlanetmathPlanetmath over a finite fieldMathworldPlanetmath of characteristic not 2, X a finite dimensional vector spaceMathworldPlanetmath over F and ϕ a regular quadratic formPlanetmathPlanetmath over X.

A regular quadratic form ϕ over X is a quadratic formMathworldPlanetmath such that for every x0 in X there is a y in X with b(x,y)0. Here b(x,y)=12(q(x+y)-q(x)-q(y)) is the associated bilinear formMathworldPlanetmathPlanetmath.

To every completion Fv of F with respect to a nontrivial valuation v we assign the vector space Xv:=FvFX and the induced quadratic form ϕv on Xv.

A quadratic form ϕ over X is an isotropic quadratic form if there is a nonzero vector xX with ϕ(x)=0.

The Hasse-Minkowski theorem can now be stated as:

Theorem 1

A regular quadratic form ϕ over a global field F is isotropic if and only if every completion ϕv is isotropic, where v runs through the nontrivial valuations of F.

The case of was first proved by Minkowski. It can be proved using the Hilbert symbol and Dirichlet’s theorem on primes in arithmetic progressions.

The general case was proved by Hasse. It can be proved using two local-global principles of class field theory, namely the Hasse norm theorem: For a cyclic field extension E/F of global fields an element aF is a norm of E/F and only if it is a norm of Ev/Fv for every valuation v of E.

and the Global square theorem: An element a of a global field F is a square if and only if it is a square in every Fv.

Title Hasse-Minkowski theorem
Canonical name HasseMinkowskiTheorem
Date of creation 2013-03-22 15:19:47
Last modified on 2013-03-22 15:19:47
Owner SirJective (9710)
Last modified by SirJective (9710)
Numerical id 8
Author SirJective (9710)
Entry type Theorem
Classification msc 15A63
Classification msc 14G05
Related topic HassePrinciple
Related topic QuadraticForm
Defines regular quadratic form