Brauer Group - Examples

Examples

• In the following cases, every finite-dimensional central division algebra over a field K is K itself, so that the Brauer group Br(K) is trivial:

• K is an algebraically closed field: more generally, this is true for any pseudo algebraically closed field or quasi-algebraically closed field.

• K is a finite field (Wedderburn's theorem);

• K is the function field of an algebraic curve over an algebraically closed field (Tsen's theorem);

• An algebraic extension of Q containing all roots of unity.

• The Brauer group Br(R) of the field R of real numbers is the cyclic group of order two. There are just two non-isomorphic real division algebras with center R: the algebra R itself and the quaternion algebra H. Since H ⊗ H ≅ M(4,R), the class of H has order two in the Brauer group. More generally, any real closed field has Brauer group of order two.

• K is complete under a discrete valuation with finite residue field. Br(K) is isomorphic to Q/Z.