, , then , showing is a well defined operation. This forms a group structure on the set of such equivalence classes called the Brauer group, denoted . Another definition is given by the torsion subgroup of the etale cohomology group which is called the cohomological Brauer group. These two definitions agree when is a field. Brauer group using Galois cohomology There is another equivalent definition of the Brauer group using Galois cohomology. For a field extension there is a cohomological Brauer group defined as and the cohomological Brauer group for is defined as where the colimit is taken over all finite Galois field extensions. Computation for a local field Over a local non-archimedean field , such as the p-adic numbers , local class field theory gives the isomorphism of abelian groups:pg 193 This is because given abelian field extensions there is a short exact sequence of Galois groups and from Local class field theory, there is the following commutative diagram: where the vertical maps are isomorphisms and the horizontal maps are injections. n-torsion for a field Recall there is the Kummer sequence giving a long exact sequence in cohomology for a field . Since Hilbert's Theorem 90 implies , there is an associated short exact sequence showing the second etale cohomology group with coefficients in the n-th roots of unity is Generators of n-torsion classes in the Brauer Group over a field The Galois symbol, or norm-residue symbol, is a map from the n-torsion Milnor K-theory group to the etale cohomology group , denoted by It comes from the composition of the cup product in etale cohomology with the Hilbert's Theorem 90 isomorphism hence It turns out this map factors through , whose class for is represented by a cyclic algebra . For the Kummer extension where , take a generator of the cyclic group, and construct . There is an alternative, yet equivalent construction through Galois cohomology and etale cohomology. Consider the short exact sequence of trivial -modules The long exact sequence yields a map For the unique character with , there is a unique lift and note the class is from the Hilberts theorem 90 map . Then, since there exists a primitive root of unity , there is also a class It turns out this is precisely the class . Because of the Norm residue isomorphism theorem, is an isomorphism and the -torsion classes in are