mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Arason invariant is a cohomological invariant associated to a

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...

of even rank and trivial

discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...

and

Clifford invariant Clifford may refer to: People *Clifford (name), an English given name and surname, includes a list of people with that name *William Kingdon Clifford * Baron Clifford *Baron Clifford of Chudleigh * Baron de Clifford *Clifford baronets * Clifford f ...

over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

''k'' of characteristic not 2, taking values in H³(''k'',Z/2Z). It was introduced by . The Rost invariant is a generalization of the Arason invariant to other algebraic groups.

Definition

Suppose that ''W''(''k'') is the Witt ring of quadratic forms over a field ''k'' and ''I'' is the ideal of forms of even dimension. The Arason invariant is a group homomorphism from ''I''³ to the Galois cohomology group H³(''k'',Z/2Z). It is determined by the property that on the 8-dimensional diagonal form with entries 1, –''a'', –''b'', ''ab'', -''c'', ''ac'', ''bc'', -''abc'' (the 3-fold Pfister form«''a'',''b'',''c''») it is given by the

cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutat ...

of the classes of ''a'', ''b'', ''c'' in H¹(''k'',Z/2Z) = ''k''*/''k''*². The Arason invariant vanishes on ''I''⁴, and it follows from the

Milnor conjecture In mathematics, the Milnor conjecture was a proposal by of a description of the Milnor K-theory (mod 2) of a general field ''F'' with characteristic different from 2, by means of the Galois (or equivalently étale) cohomology of ''F'' wi ...

proved by Voevodsky that it is an

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...

from ''I''³/''I''⁴ to H³(''k'',Z/2Z).

References

* * * * Algebraic groups {{algebra-stub