Albert–Brauer–Hasse–Noether theorem

Remarks

The proof uses class field theory: one reduces to cyclic algebras and then uses the Hasse norm theorem. That theorem says that in a cyclic extension of global fields, an element that is a local norm everywhere is a global norm. Example: the quaternion algebra $\left(\frac{-1, -1}{\Q}\right)$ splits over $\Q_p$ for every odd prime $p$, but not over $\Q_2$ or over $\R$. So it is not split over $\Q$, in agreement with the theorem. Class field theory gives more: the sequence $0 \to \mathrm{Br}(K) \to \bigoplus_v \mathrm{Br}(K_v) \to \Q/\Z \to 0$, with the sum of local invariants as the last map, is exact. Because of this sum rule it is enough to check all places but one, and a quaternion algebra over $K$ is ramified at an even number of places.

Going deeper

For a student

The theorem is a cornerstone of the arithmetic of central simple algebras over number fields. With the exact sequence above it gives the classification of central simple algebras over a global field by their local invariants. Consequences are that every central simple algebra over a global field is cyclic and that its period equals its index. For quaternion algebras it reduces to the Hasse–Minkowski theorem for the ternary quadratic form $a x^2 + b y^2 - a b z^2$.