# 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.

## Local fields

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**.^{[1]}

There is a commutative subfield *L* of *D* which is unramified over *K*, and *D* splits over *L*.^{[2]} 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*.^{[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,π^{k}) for some *k* mod *n*, where φ is the Frobenius map and π is a uniformiser.^{[6]} The invariant map attaches the element *k*/*n* mod 1 to the class. This exhibits the invariant map as a homomophism

The invariant map extends to Br(*K*) by representing each class by some element of Br(*L*/*K*) as above.^{[3]}^{[4]}

For a non-Archimedean local field, the invariant map is a group isomorphism.^{[3]}^{[7]}

In the case of the field **R** of real numbers, there are two Brauer classes, represented by the algebra **R** itself and the quaternion algebra **H**.^{[8]} It is convenient to assign invariant zero to the class of **R** and invariant 1/2 modulo 1 to the quaternion class.

In the case of the field **C** of complex numbers, the only Brauer class is the trivial one, with invariant zero.^{[9]}

## Global fields

For a global field *K*, given a central simple algebra *D* over *K* then for each valuation *v* of *K* we can consider the extension of scalars *D*_{v} = *D* ⊗ *K*_{v} The extension *D*_{v} splits for all but finitely many *v*, so that the *local invariant* of *D*_{v} is almost always 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. The injectivity of the left arrow is the content of the Albert–Brauer–Hasse–Noether theorem. Exactness in the middle term is a deep fact from global class field theory.

## References

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## Further reading

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