Hasse–Minkowski theorem

Remarks

The direction "only if" is clear; the content is that local zeros give a global zero. The proof goes by the number of variables, with the Hilbert symbol as the main tool. The case $n = 3$ is essentially Legendre's theorem on $ax^2 + by^2 + cz^2 = 0$; the case $n = 4$ uses Dirichlet's theorem on arithmetic progressions, and the case $n \ge 5$ reduces to $n = 4$. The theorem holds over every number field, with its completions in place of $\R$ and the $\Q_p$. Example: $x^2 + y^2 - 3z^2$ has the real zero $(\sqrt{3}, 0, 1)$, but no nonzero zero in $\Q_3^3$; hence it has no nonzero rational zero. A consequence: a nonzero $a \in \Q$ is represented by $Q$ over $\Q$ if and only if it is represented over $\R$ and over every $\Q_p$. Another: two quadratic forms over $\Q$ are equivalent over $\Q$ if and only if they are equivalent over $\R$ and over every $\Q_p$. The restriction to quadratic forms is necessary: Selmer's cubic $3x^3 + 4y^3 + 5z^3$ has nonzero zeros over $\R$ and every $\Q_p$, but none over $\Q$.

Going deeper

For a student

This is the main example of the Hasse principle. In practice one computes Hilbert symbols $(a, b)_p$ at the primes dividing $2$ and the coefficients; at all other primes a form of rank at least $3$ is automatically isotropic. The theorem is the basis of the classification of quadratic forms over $\Q$ by discriminant, signature and Hasse invariants. For cubic and higher degree equations the principle can fail, and the study of the failure leads to the Tate–Shafarevich group of elliptic curves.