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 invariant plays a role in local class field theory.
Let K be a local field with valuation v and D a K-algebra. We may assume D is a division algebra with centre K of degree n. The valuation v can be extended to D, for example by extending it compatibly to each commutative subfield of D: the value group of this valuation is (1/n)Z.
There is a commutative subfield L of D which is unramified over K, and D splits over L. The field L is not unique but all such extensions are conjugate by the Skolem–Noether theorem, which further shows that any automorphism of L is induced by a conjugation in D. Take γ in D such that conjugation by γ induces the Frobenius automorphism of L/K and let v(γ) = k/n. Then k/n modulo 1 is the Hasse invariant of D. It depends only on the Brauer class of D.
The Hasse invariant is thus a map defined on 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,πk) 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. This exhibits the invariant map as a homomorphism
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