Hasse-Minkowski theorem
The Hasse-Minkowski theorem is a classical example of the Hasse principle.
Let be a global field, i.e. a number field or a rational function field over a finite field of characteristic not , a finite dimensional vector space over and a regular quadratic form over .
A regular quadratic form over is a quadratic form such that for every in there is a in with . Here is the associated bilinear form.
To every completion of with respect to a nontrivial valuation we assign the vector space and the induced quadratic form on .
A quadratic form over is an isotropic quadratic form if there is a nonzero vector with .
The Hasse-Minkowski theorem can now be stated as:
Theorem 1
A regular quadratic form over a global field is isotropic if and only if every completion is isotropic, where runs through the nontrivial valuations of .
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 of global fields an element is a norm of and only if it is a norm of for every valuation of .
and the Global square theorem: An element of a global field is a square if and only if it is a square in every .
Title | Hasse-Minkowski theorem |
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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 |