Then k/n modulo 1 is the Hasse invariant of D. It depends only on the Brauer class of D.[3] The Hasse invariant is thus a map defined on the Brauer group of a local field K to the divisible group Q/Z.
[3][4] 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,[5] which by the Grunwald–Wang theorem and the Albert–Brauer–Hasse–Noether theorem we may take to be a cyclic algebra (L,φ,πk) for some k mod n, where φ is the Frobenius map and π is a uniformiser.
In the case of the field C of complex numbers, the only Brauer class is the trivial one, with invariant zero.
The Brauer group Br(K) fits into an exact sequence[8][9] where S is the set of all valuations of K and the right arrow is the sum of the local invariants.
Exactness in the middle term is a deep fact from global class field theory.