Albert–Brauer–Hasse–Noether theorem

The theorem embeds the Brauer group of a number field into the direct sum of the Brauer groups of its completions, identifying its image with tuples whose local invariants sum to zero.

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Brauer group: The group of equivalence classes of central simple algebras over a field, with tensor product as its operation. The theorem describes the Brauer group of a number field through the groups of its completions.
Number field: A finite extension of the rational numbers. The theorem's global field is a number field.
Richard Brauer: A German-American mathematician whose work shaped representation theory and the theory of central simple algebras. The theorem bears Brauer's name and grew from his work on algebras.
Quaternion algebra: A four-dimensional central simple algebra generated by two elements whose squares and products satisfy specified relations. Its local invariants determine whether a quaternion algebra over a number field splits globally.
Brauer–Hasse–Noether exact sequence: The exact-sequence formulation relating a number field's Brauer group, its local Brauer groups, and rational invariants. It states the theorem's maps and kernel condition explicitly.
Local invariant: A homomorphism from the Brauer group of a local field to a subgroup of the rational numbers modulo integers. These values encode each completed algebra and must sum to zero globally.
Place (mathematics): An equivalence class of absolute values on a field, encoding an Archimedean or non-Archimedean notion of size. Each place determines one completion included in the theorem.
Helmut Hasse: A German mathematician known for major contributions to algebraic number theory and local-global methods. Hasse established the local-global framework underlying the theorem.
Hasse principle: A local-global principle asserting that a global object exists when compatible objects exist over all completions. The theorem gives a precise local-global statement for Brauer classes, including its obstruction.
Tsen's theorem: A theorem stating that a function field in one variable over an algebraically closed field has trivial Brauer group. It contrasts number fields, whose Brauer groups can be nontrivial but are constrained by local invariants.