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:
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 .
|Date of creation||2013-03-22 15:19:47|
|Last modified on||2013-03-22 15:19:47|
|Last modified by||SirJective (9710)|
|Defines||regular quadratic form|