In mathematics, the Hasse invariant of an algebra is an invariant attached to a

Brauer class Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik Br ...

algebras over a field In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition a ...

. The concept is named after

Helmut Hasse Helmut Hasse (; 25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory and ...

. The invariant plays a role in

local class field theory In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite re ...

Local fields

Let ''K'' be a

local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact t ...

with valuation ''v'' and ''D'' a ''K''-algebra. We may assume ''D'' is a

division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a fie ...

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.Serre (1967) p.137 There is a commutative subfield ''L'' of ''D'' which is unramified over ''K'', and ''D'' splits over ''L''.Serre (1967) pp.130,138 The field ''L'' is not unique but all such extensions are conjugate by the

Skolem–Noether theorem In ring theory, a branch of mathematics, the Skolem–Noether theorem characterizes the automorphisms of simple rings. It is a fundamental result in the theory of central simple algebras. The theorem was first published by Thoralf Skolem in 1927 in ...

, 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''.Serre (1967) p.138 The Hasse invariant is thus a map defined on the

Brauer group Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik ...

of a

''K'' to the

divisible group In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an ''n''th multiple for each positive in ...

Q/Z.Lorenz (2008) p.232 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'',Lorenz (2008) pp.225–226 which by the

Grunwald–Wang theorem In algebraic number theory, the Grunwald–Wang theorem is a local-global principle stating that—except in some precisely defined cases—an element ''x'' in a number field ''K'' is an ''n''th power in ''K'' if it is an ''n''th power in the comp ...

and the

Albert–Brauer–Hasse–Noether theorem In algebraic number theory, the Albert–Brauer–Hasse–Noether theorem states that a central simple algebra over an algebraic number field ''K'' which splits over every completion ''K'v'' is a matrix algebra over ''K''. The theorem is an e ...

we may take to be a

cyclic algebra In algebra, a cyclic division algebra is one of the basic examples of a division algebra over a field, and plays a key role in the theory of central simple algebras. Definition Let ''A'' be a finite-dimensional central simple algebra over a fiel ...

(''L'',φ,π^''k'') for some ''k'' mod ''n'', where φ is the

Frobenius map In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism m ...

and π is a uniformiser.Lorenz (2008) p.226 The invariant map attaches the element ''k''/''n'' mod 1 to the class. This exhibits the invariant map as a homomorphism :

\underset : \operatorname(L/K) \rightarrow \mathbb/\mathbb .

The invariant map 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 In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two grou ...

.Lorenz (2008) p.233 In the case of the field R of

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

s, there are two Brauer classes, represented by the algebra R itself and the

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...

algebra H.Serre (1979) p.163 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.

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 An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context o ...

Gille & Szamuely (2006) p.159 :

0\rightarrow \textrm(K)\rightarrow \bigoplus_ \textrm(K_v)\rightarrow \mathbf/\mathbf \rightarrow 0,

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

. Exactness in the middle term is a deep fact from

global class field theory In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit ...

References

* * * *

Local fields

Global fields

References

Further reading