Hasse invariant of an algebra

In mathematics, the Hasse invariant of an algebra is an invariant attached to a Brauer class of algebras over a field. The concept is named after Helmut Hasse.

The Hasse invariant is a map from the Brauer group of a local field K to the divisible group Q/Z. Every class in the Brauer group is represented by a class in the Brauer group of an unramified extension of L/K of degree n, which by the Grunwald–Wang theorem and the Albert–Brauer–Hasse–Noether theorem we may take to be a cyclic algebra (L,φ,π^k) for some k mod n, where φ is the Frobenius map and π is a uniformiser. The invariant map attaches the element k/n mod 1 to the class and gives rise to a homomophism

${\underset {L/K}{\operatorname {inv} }}:\operatorname {Br} (L/K)\rightarrow \mathbb {Q} /\mathbb {Z} .$

The invariant map now extends to Br(K) by representing each class by some element of Br(L/K) as above.

For a non-Archimedean local field, the invariant map is a group isomorphism.