# Hasse-Minkowski theorem

Let $F$ be a global field  , i.e. a number field  or a rational function field  over a finite field  of characteristic not $2$, $X$ a finite dimensional vector space  over $F$ and $\phi$ a regular quadratic form  over $X$.

A regular quadratic form $\phi$ over $X$ is a quadratic form  such that for every $x\neq 0$ in $X$ there is a $y$ in $X$ with $b(x,y)\neq 0$. Here $b(x,y)=\frac{1}{2}(q(x+y)-q(x)-q(y))$ is the associated bilinear form   .

To every completion $F_{v}$ of $F$ with respect to a nontrivial valuation $v$ we assign the vector space $X_{v}:=F_{v}\otimes_{F}X$ and the induced quadratic form $\phi_{v}$ on $X_{v}$.

A quadratic form $\phi$ over $X$ is an isotropic quadratic form if there is a nonzero vector $x\in X$ with $\phi(x)=0$.

The Hasse-Minkowski theorem can now be stated as:

###### Theorem 1

A regular quadratic form $\phi$ over a global field $F$ is isotropic if and only if every completion $\phi_{v}$ is isotropic, where $v$ runs through the nontrivial valuations of $F$.

The case of $\mathbb{Q}$ 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 $a\in F$ is a norm of $E/F$ and only if it is a norm of $E_{v}/F_{v}$ 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 $F_{v}$.

Title Hasse-Minkowski theorem HasseMinkowskiTheorem 2013-03-22 15:19:47 2013-03-22 15:19:47 SirJective (9710) SirJective (9710) 8 SirJective (9710) Theorem msc 15A63 msc 14G05 HassePrinciple QuadraticForm regular quadratic form